text:
305 E. Capitol St. | Jackson, MS 39201
P.O. Box 1366 | Jackson, MS 39215-1366
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VESTRY NOMINEES
Vestry Nominees 2022
For Warden
[Jump to Vestry]
Holli Roach
Standing for election to be Senior Warden is not a call I take lightly. It is a privilege to even be asked to give it prayerful consideration to be listed on the ballot. As a nurse for 34 years I have been inspired by the Holy acts I see at work, at Church, and at random times. Over the past 37 years at St. Andrew's I have been an EYC leader, taught Junior/Senior High Sunday School, helped coordinate and volunteer with Vacation Bible study, lead the Parrish Life Committee to coordinate Wednesday evenings and Summer Sundays, served on a Discernment Committee, assisted with Annual Giving campaigns, Co-chaired the events to welcome our most recent Dean, I am a part of Lay Eucharistic Ministry, and served as a Vestry member to fulfill VA Patterson's term.
St. Andrew's is my home and a place for renewal each time I walk on these hallowed grounds. So much of my life since moving to Jackson in 1984 has been centered in St. Andrew's. Our premarital counseling was done here although we married in Atlanta where I was raised. Dan and I have Christened and Confirmed all four of our children in the Nave. We've eulogized three of our parents here when they died. We have lived through the good times and challenging times of our Cathedral. I consider it an honor to pay it forward as Senior Warden and help welcome more new people to our home while sustaining those who are still here. I would also like to help us continue to grow our Community partnerships in and around the Jackson metro area. What has transpired these past two years with COVID has inspired me to no longer be reluctant about being more involved with our Church Family. If not now, then when? Now continues to be an exciting time to imagine the unimaginable. As we being to revisit the familiarity of in person worship, communion at the Altar, we are also almost giddy with childlike excitement of what can we make new.
Robert Langford
St. Andrew's Cathedral has been a foundational part of my life, shaping my understanding of the world since I was baptized in our nave in 1964. Our Cathedral community has taught me about the blessings and burdens of hearing, discerning and heeding God's call to follow the example of Christ in my relationships with others, my family, my parish community, and out in the larger world. This church and its community have served as my spiritual touchstone for all my 58 years. My wife Betsy Bradley and I met at a church event and were married here, and our children Martha and William were baptized and confirmed here. I'm the executive director of Operation Shoestring, a nonprofit ministry, where I've worked since 1995.
I've served on the Vestry twice before, most recently when Anne was called to be our Dean, concluding my last term in January 2021. Additionally, I've served as a chalice bearer and am currently a member of the Community Partnership Commission and the Finance Committee.
In discerning God's call to serve St. Andrew's, it's my hope that if selected as your senior warden I can use my decades of organizational management experience and my love of this community and our church to help us all more fully know God's call for us, both at St. Andrew's and in our larger community. This is a dynamic time for St. Andrew's as we navigate the new normal brought on by the pandemic and the evolving state of religion in America. It would be an honor to use whatever talents and faith I possess to help our Dean, Vestry and the rest of our parish continue to move our beloved St. Andrew's into the future as we stay firmly grounded in the Anglican tradition.
For Vestry
[Jump to Warden]
Jump to Vestry Nominee:
Sheryl Fox | Alexandra Long | Jimmy Nix | Crystal Secoy
Lucien Smith | Stephen Stray | Dorothy Triplett
Sheryl Fox
Over the last several years I have become more and more involved with activities and committees/ guilds at St. Andrews. This has allowed me to meet more of my fellow parishioners, and has allowed me to develop very deep and meaningful friendships. This sharing with others has also encouraged me to use my seeking to deepen my personal spiritual growth.
I have been a member of the Episcopal Church for over 40 years, and I have been a communicant at St. Andrew's for 21. I currently serve as Lay Eucharistic Visitor, a member of the flower guild, a member of St. Michael's guild, and as a greeter/connector. I am a co-chair for the connectors on the Invite Welcome Connect group.
I am the Meals on Wheels coordinator for St. Andrew's, Galloway, and Beth Israel. I participate in the Stewpot ministries, and I am a current 3rd year class member of Education for Ministry.
I would be honored to be given the opportunity to serve on the vestry.
Alexandra Long
St. Andrew's is a place that I love to serve in any way I'm needed, because it has given me a home and place to grow in my faith. I was raised in the Episcopal Church and joined the Cathedral after college. I was married here in 2012, my father was buried here in 2017, and my youngest daughter was baptized here in 2019. The church has carried me in biggest days of my life as well as the days in between. I have served on the Guild of the Child, helped with VBS and EYC and now as co-chair of Altar Guild. Before moving back to Jackson in 2018, I served as assistant youth leader and helped create an evening service at Christ Church Cathedral in Nashville, TN. Since moving home I have started to work for Garraway Browne and Company part-time and raising Elizabeth and Mary Stuart with my husband David Long.
Jimmy Nix
I have been a part of St. Andrews Cathedral for my entire life. I was baptized at the Church and it's the only church community I have ever known. This church is a central part of my life and the foundation for my relationship with God. I love this church and believe this church is a staple in the community of Jackson. We need this church to continue to grow. We are a wonderful church. I have served breakfast on Tuesdays, been a part of the Dean's List, and cooked with the holy smokers for years. My expertise is a simple one. I am a people person. I would bring the passion of growing our church by getting young families involved in the cathedral. We are a great church people just need to realize how great we are. Regardless of how this plays out I thank yall for the consideration.
Crystal Utley Secoy
First, thank you to the Vestry Nominating Committee for nominating me. My personal mission statement is to lead, restore, and empower others in a proactive, positive, and biblically-centered manner through collaboration, problem-solving, informed strategic decision-making, and candor. It would be an honor to unite my mission with the mission of our Cathedral in the City to empower, serve, and transform hearts for God as a vestry member.
As a graduate of Saint Andrew's Episcopal School (1st – 12th,) I had the opportunity to attend services at the Cathedral, and, through my education, I learned the importance of tolerance, embracing other cultures, and being an open-minded person. Growing up, my family regularly attended Christ United Methodist Church. At CUMC while in law school, I led Good News Clubs at the Church's after school care on West Street and served on their committee supervising ministry and missions.
As an attorney, I started my practice in Charleston, South Carolina. However, in the wake of Hurricane Katrina, God led me to come home and to manage disaster-related legal aid in partnership with Lutheran Episcopal Services and other nonprofits. While serving as Pro Bono Counsel for the Mississippi Center for Justice, I received the Spirit of Service Award from the Corporation for National and Community Service and awards from Governor Haley Barbour and the American Red Cross. As the current Director of Consumer Protection for the Mississippi Attorney General and in my prior role, supervising hundreds of volunteer attorneys and law students, I have gained leadership experience that I can utilize to the benefit of our Cathedral.
Being on the Coast also gave me the chance to worship at St. John's Episcopal Church of Ocean Springs. Fast forward ten years, and my husband, Nick, and I transferred our membership to St. Andrew's from Chapel of the Cross Rolling Fork (his hometown) in 2016, largely due to St. Andrew's community of young families and the Cathedral's activities for small children. I began to have a clear sense that I wanted to be a part of supporting this congregation. Presently, I lead the Cathedral's Guild of the Christ Child, serve as a lector and a litanist, and participate on the Children and Youth Commission as we have worked together to re-evaluate the Cathedral's youth formation and activities. I previously served on the Parish Life Commission and continue to volunteer for events.
I believe it is crucial to have strong communication within our congregation and that actions of, or on behalf of, the Cathedral as a whole are thoughtfully addressed by the vestry before proceeding. In this era, it is particularly important to make sure that we are led by God, focused on the love of God and neighbor rather than on one constituency to the alienation of another.
Lastly, and most importantly, as the mom of James (almost seven) and Jude (four,) I have a keen sense of how the future of our church and of our community lies within our children and the importance of nurturing, protecting and raising them in Christ. I'm very grateful to the Cathedral for being there for James and Jude - from our nursery team to Vacation Bible School to dressing up like the wise men!
Lucien Smith
I have been a member of the parish since my confirmation in 2009, but have considered Saint Andrew's home for much of my life. Having gone to Saint Andrew's Episcopal School from prekindergarten through my graduation in 1999, many of my memories of childhood involve Friday chapels or major events hosted at the cathedral. I made my membership official after participating in the 2009 Inquirers' Class and was confirmed that May, having been raised in the Methodist faith. I am an attorney with Balch & Bingham LLP here downtown and live in Jackson with my wife Beth and our children Liam (5), McEwen (3), and Cooper (9 months). Ours is a special community, of which I am proud to be a member, and over the previous several months, I have felt an increasing desire to serve it. I hope that serving on the vestry would be one of several ways I can serve this wonderful place and community of believers.
Stephen Stray
St Andrew's Cathedral has been my church home since 2006. Over the last 31 years, it has been my joy and honor to serve at multiple levels of the Episcopal Church. At the Parish level, I have served on Vestry (St. Andrew's Cathedral 2016-19, St Paul's Oklahoma City [3 years]) and on the former Adult Formation Commission, which I also chaired for some years. I also co-chaired the Stewardship Campaigns in 2018 and 2019. At the Diocesan level, I have served on both the Executive Committee (5 years) and Standing Committee (4 years), serving as President of the standing Committee in 2020, one of the most challenging years in the history of the Diocese. At the national level, I served on the Executive Council Advisory Committee on Science, Technology, and Faith (6 years). I have been a member of both of St Andrew's fine choirs, have taught Sunday School, and participated in many other church activities. In my work life, I am a researcher and teacher at the University of Mississippi Medical Center, where I have helped to train nearly 3000 physicians and dentists.
More important than what I have contributed to St. Andrew's is what I have taken from it- St Andrew's has been my spiritual home and social community for more than 15 years. In that time, I have laughed and cried with my St Andrew's family, sharing the joys and sorrows that we have experienced individually and corporately. As a parish, we have been through some difficult times, but thanks to the love of people in the pews for this place, and capable leadership from both our lay and clergy leaders, we remain strong as we look toward the end of our second century in the heart of Jackson- and beyond. Like all "mainstream" congregations, we face a future that is very different from the past we have known, but I have confidence that we will continue to live into our mission of being the "Cathedral in the City". If elected to vestry, I expect to work closely with Dean Maxwell and other clergy and lay leaders to continue to live into our vision of being the hands, the feet, and the body of Christ.
Dorothy Triplett
I am grateful to have been asked to place my name in nomination for the Vestry of St. Andrew's, where I have been a member since moving to Jackson in 1987. Prior to that, I was a member of St. Francis Mission in Philadelphia MS, and St. Mark's in Milwaukee, WI.
At this time in my spiritual journey, I have the time, energy, and willingness to serve as a member of the Vestry – and hopefully can bring a viewpoint of both age and community advocacy. My extensive experience with non-profits, both as a volunteer and a staff member, has given me an opportunity to explore and examine how we can expand our role as "The Cathedral in the City" and our participation and visibility in the community as a whole and in the interfaith work being done in the metro area. It's so important for us as a congregation and as members of this caring Christian community to see how we can truly listen and learn from each other – and embrace our differences and further examine our culture and learn how to better be truly inclusive of all of our brothers and sisters.
That same spiritual journey has been enhanced by every relationship I have with fellow members, clergy and staff of the St. Andrew's community and all aspects of work and worship here. My life has been enriched by them, and I believe I have grown in my understanding of faith and my responsibilities as a member of what Presiding Bishop Michael Curry calls the "Episcopal expression of the Jesus movement."
St. Andrew's—Member of: Parish Choir, Community Partnership Commission (liaison for CONTACT the Crisis Line); Invite, Welcome, Connect group; Holy Smokers/Smokettes. Lector/Litanist; Parish Visitor; just beginning to serve at St. Catherine's as a lay reader; led Noonday Prayers online during pandemic; Front Desk receptionist; represent St. Andrew's on Leadership Team of Working Together Jackson and Working Together Mississippi; Volunteer tutor at Rowan School and Arise2Read; and volunteer whenever asked or needed.
Episcopal Diocese of MS: Past member of Standing Committee; Past member of Commission on Aging; Past member of Racial Reconciliation Commission; Licensed Lay reader.
Church activities/responsibilities prior to coming to St. Andrew's:
Member of the Standing Committee on Church in Metropolitan Areas for the National Episcopal Church in the 1970's.
Mark's, Milwaukee, WI: Vestry, choir, lay reader, Chair of Strawberry Festival (art fair), active in all usual parish events.
Francis, Philadelphia, MS: Member of Bishop's Committee, Licensed Layreader
Relevant Volunteer work: 18 years as a Certified Telephone Crisis Counselor for CONTACT the Crisis Line – (prior to 13 years on staff) – and current Board member; Member of Executive Committee for Working Together Jackson; Board member of New Way Mississippi Inc. – program for persons leaving prison – provides transitional housing, counseling, job search assistance; Former Board member of Jackson 2000; Graduate of Leadership Jackson and former President of the Leadership Jackson Alumni; plus numerous other roles for various organizations.
Relevant Work experience (Partial list): Deputy Director of Project Equality of Wisconsin; Education Director of Winston County (MS) HeadStart; Planner/grant writer, Prairie Opportunity Community Action Agency (Starkville MS); Director of statewide federally-funded Fixed Income Community Counseling; Deputy Director of the (MS) Governor's Office of Federal/State Programs; Business Services Director and Director of Constituent Services-Office of the MS Secretary of State; Public Affairs and Communications Director- office of Jackson Mayor Harvey Johnson; Director of Training and Community Services for CONTACT the Crisis Line.
305 E. Capitol St. I Jackson, MS 39201


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\section{Introduction}\label{sec:1}
We assume that the reader is familiar with the basic theory of strongly regular graphs and difference sets. For strongly regular graphs (srgs), our main references are \cite{BH} and \cite{cg}. For difference sets, we refer the reader to \cite{Lander} and Chapter 6 of \cite{bjl}. Strongly regular graphs are closely related to many other combinatorial/geometric objects, such as two-weight codes, two-intersection sets, $m$-ovoids, $i$-tight sets, and partial difference sets. For these connections, we refer the reader to \cite[p.~132]{BH}, \cite{CK86, Ma}, and some more recent papers \cite{FMX, DDMR, BLMX} on Cameron-Liebler line classes and hemisystems.

Let $\Gamma$ be a (simple, undirected) graph. The adjacency matrix of $\Gamma$ is the $(0,1)$-matrix $A$ with both rows and columns indexed by the vertex set of $\Gamma$, where $A_{xy} = 1$ when there is an edge between $x$ and $y$ in $\Gamma$ and $A_{xy} = 0$ otherwise. A useful way to check whether a graph is strongly regular is by using the eigenvalues of its adjacency matrix. For convenience we call an eigenvalue {\it restricted} if it has an eigenvector perpendicular to the all-ones vector ${\bf 1}$. (For a $k$-regular connected graph, the restricted eigenvalues are the eigenvalues different from $k$.)

\begin{theorem}\label{char}
For a simple graph $\Gamma$ of order $v$, not complete or edgeless, with adjacency matrix $A$, the following are equivalent:
\begin{enumerate}
\item $\Gamma$ is strongly regular with parameters $(v, k, \lambda, \mu)$ for certain integers $k, \lambda, \mu$,
\item $A^2 =(\lambda-\mu)A+(k-\mu) I+\mu J$ for certain real numbers $k,\lambda, \mu$, where $I, J$ are the identity matrix and the all-ones matrix, respectively,
\item $A$ has precisely two distinct restricted eigenvalues.
\end{enumerate}
\end{theorem}

For a proof of Theorem~\ref{char}, we refer the reader to \cite{BH}. An effective method to construct strongly regular graphs is by using Cayley graphs. Let $G$ be an additively written group of order $v$, and let $D$ be a subset of $G$ such that $0\not\in D$ and $-D=D$, where $-D=\{-d\mid d\in D\}$. The {\it Cayley graph over $G$ with connection set $D$}, denoted ${\rm Cay}(G,D)$, is the graph with the elements of $G$ as vertices; two vertices are adjacent if and only if their difference belongs to $D$. In the case when $\Cay(G,D)$ is a strongly regular graph, the connection set $D$ is called a (regular) {\it partial difference set}. Examples of strongly regular Cayley graphs are the Paley graphs ${\rm P}(q)$, where $q$ is a prime power congruent to 1 modulo 4, the Clebsch graph, and the affine orthogonal graphs (\cite{BH}). For $\Gamma={\rm Cay}(G,D)$ with $G$ abelian, the eigenvalues of $\Gamma$ are exactly $\chi(D):=\sum_{d\in D}\chi(d)$, where $\chi$ runs through the character group of  $G$. This fact reduces the problem of computing eigenvalues of abelian Cayley graphs to that of computing some character sums, and is the underlying reason why the Cayley graph construction has been very effective for the purpose of constructing srgs. The survey of Ma~\cite{Ma} contains much of what is known about partial difference sets and about connections with strongly regular graphs.

A $(v,k,\lambda,\mu)$ srg is said to be of
{\em{Latin square type}} (respectively, {\it negative Latin square
type}) if $(v,k,\lambda,\mu) = (n^2, r(n-\epsilon), \epsilon n+r^2-3
\epsilon r, r^2 - \epsilon r)$ and $\epsilon = 1$ (respectively, $\epsilon=-1$). When $v$ (the number of vertices) is a prime power, many constructions of srgs with Latin square or negative Latin square type parameters are known. For example, the srgs arising from partial spreads of $\PG(2m -1,q)$ have Latin square parameters, and the affine orthogonal graphs, ${\rm VO}^{-}(2m,q)$, have negative Latin square type parameters. Still the range of $r$ in the parameters $(n^2, r(n-\epsilon), \epsilon n+r^2-3
\epsilon r, r^2 - \epsilon r)$ of the known srgs of Latin square or negative Latin square type can sometimes be limited; moreover Latin square and negative Latin square type strongly regular Cayley graphs with certain extra properties\footnote{For example, the elements of the connection set must all lie on a quadratic surface.} have found many connections with finite geometric objects such as $m$-ovoids and $i$-tight sets (cf. \cite{FMX, DDMR, BLMX}). Therefore it is of interest to construct more strongly regular Cayley graphs of Latin square or negative Latin square type. The purpose of the current paper is two fold. First, we give new constructions of strongly regular Cayley graphs, and obtain some new srgs. Secondly, we unify and give simpler proofs for some recent constructions of strongly regular Cayley graphs.

The paper is organized as follows. In Section 2, we review some basic properties of Gauss sums which will be used in later sections. In Section 3, we give two constructions of strongly regular Cayley graphs on the additive group of $\F_{q^2}$ by lifting a cyclotomic strongly regular graph on $\F_q$. The first lifting construction (Proposition~\ref{theorem:main1}) is of ``elliptic" type, and it was already given in \cite{MX}. The second lifting construction (Proposition~\ref{theorem:main2}) is of ``hyperbolic" type, and this constrution is new. In Section 4, we
generalize and unify the constructions of strongly regular Cayley graphs corresponding to $m$-ovoids and $i$-tight sets in \cite{FMX, BLMX}. We give a general construction of strongly regular Cayley graphs by using a certain partition of a subdifference set (and its complement) of the Singer difference set. When the subdifference sets arise from subfields, we recover the results in \cite{FMX, BLMX}. In Sections 5, 6, and 7, we apply the general construction in Section 4 to the three known cases of subdifference sets of the Singer difference sets, namely, the semiprimitive case, the sporadic case, and the subfield case. We either recover strongly regular Cayley graphs constructed in some of our recent papers \cite{Mo, FMX, BLMX}, or we produce new strongly regular Cayley graphs. In particular, Corollaries 7.7 and 7.9 give strongly regular Cayley graphs with the same parameters as the affine polar graphs. By using a computer, it is shown that the newly constructed graphs in Corollaries 7.7 and 7.9 are not isomorphic to the affine polar graphs when the parameters are small.

\section{Preliminaries}
We will use Gauss sums and Gauss periods to compute character values of certain subsets of  $\F_{q}$, the finite field of order $q$. So it is helpful to introduce characters of both kinds of finite fields, and review basic properties of Gauss sums. Let $p$ be a prime, $f$ a positive integer, and $q=p^f$. The canonical additive character $\psi_{\F_q}$ of $\F_q$ is defined by
$$\psi_{\F_q}\colon\F_q\to \C^{\ast},\qquad\psi_{\F_q}(x)=\zeta_p^{\tr _{q/p}(x)},$$
where $\zeta_p={\rm exp}(\frac {2\pi i}{p})$ is a complex primitive $p$-th root of unity and $\Tr _{q/p}$ is the trace function from $\F_q$ to $\F_p$
defined by
$\tr_{q/p}(x) = x + x^p + x^{p^2} +\cdots + x^{p^{f-1}}$.
All the additive characters of $\F_q$ can be obtained from the canonical one. For $a\in \F_q$, define
\begin{equation}\label{additive}
\psi_a(x)=\psi_{\F_q}(ax), \;\forall x\in \F_q.
\end{equation}
Then $\{\psi_a\mid a\in \F_q\}$ is the group of additive characters of $\F_q$. For a multiplicative character
$\chi$ and the canonical
additive character $\psi_{\F_q}$ of $\F_q$, define the {\it Gauss sum} by
\[
G_q(\chi)=\sum_{x\in \F_q^\ast}\chi(x)\psi_{\F_q}(x).
\]

Some basic properties of Gauss sums are listed below:
\begin{proposition}{\em (\cite[Theorem 5.2]{LN97})}
Let $\chi$ be a multiplicative character of $\F_{q}$. Then, the following hold:
\begin{enumerate}
\item[(i)] $G_q(\chi)\overline{G_q(\chi)}=q$ if $\chi$ is nontrivial;
\item[(ii)] $G_q(\chi^p)=G_q(\chi)$, where $p$ is the characteristic of $\F_q$;
\item[(iii)] $G_q(\chi^{-1})=\chi(-1)\overline{G_q(\chi)}$;
\item[(iv)] $G_q(\chi)=-1$ if $\chi$ is trivial.
\end{enumerate}
\end{proposition}
Let $\omega$ be a fixed primitive element of $\F_q$ and $N$ a positive integer dividing $q-1$. For $0\le i\le N-1$ we set $C_i^{(N,q)}=\omega^i C_0$, where $C_0$ is the subgroup of index $N$ of $\F_q^\ast$. The {\it Gauss periods} associated with these cosets are defined by $\psi_{\F_q}(C_i^{(N,q)}):=\sum_{x\in C_i^{(N,q)}}\psi_{\F_q}(x)$, $0\le i\le N-1$, where $\psi_{\F_q}$ is the canonical additive character of $\F_q$. By orthogonality of characters, the Gauss periods can be expressed as a linear combination of Gauss sums:
\begin{equation}
\psi_{\F_q}(C_i^{(N,q)})=\frac{1}{N}\sum_{j=0}^{N-1}G_q(\chi^{j})\chi^{-j}(\omega^i), \; 0\le i\le N-1,
\end{equation}
where $\chi$ is any fixed multiplicative character of order $N$ of $\F_q$.
For example, if $N=2$,
we have
\begin{equation}\label{eq:Gaussquad}
\psi_{\F_q}(C_i^{(2,q)})=\frac{-1+(-1)^iG_q(\eta)}{2},\; 0\le i\le 1,
\end{equation}
where $\eta$ is the quadratic character of $\F_q$.

The quadratic Gauss sum, $G_q(\eta)$, can be evaluated explicitly.
\begin{theorem}\cite[Theorem~5.15]{LN97} \label{thm:Gauss}
Let $q=p^s$ be a prime power with $p$ a prime and $\eta$ be the quadratic character of $\F_q$.
Then,
\begin{equation}\label{eq:Gaussquad1}
G_q(\eta)=\begin{cases}
(-1)^{s-1}q^{1/2}& \text{ if }  p\equiv 1\,(\mod{4}), \\
(-1)^{s-1}i^s q^{1/2} & \text{ if } p\equiv 3\,(\mod{4}).
\end{cases}
\end{equation}
\end{theorem}
Also, in the semi-primitive case, the
Gauss sum can be computed.
\begin{theorem}\label{thm:semiprim}{\em (\cite[Theorem~11.6.3]{BEW97})}
Let $p$ be a prime.
Suppose that $N>2$ and $p$ is semi-primitive modulo $N$,
i.e., there exists a positive integer $j$ such that  $p^j\equiv -1\pmod{N}$. Choose
$j$ minimal and write
$f=2js$ for any positive integer $s$. Let $\chi$ be a multiplicative character of order $N$ of $\F_{p^f}$.
Then,
\[
p^{-f/2}G_{p^f}(\chi)=
\left\{
\begin{array}{ll}
(-1)^{s-1},&  \mbox{if $p=2$,}\\
(-1)^{s-1+(p^j+1)s/N},&  \mbox{if $p>2$. }
 \end{array}
\right.
\]
\end{theorem}

The following theorems are referred to as the  {\it Davenport-Hasse lifting formula} and the  {\it Davenport-Hasse product formula}, respectively.
\begin{theorem}\label{thm:lift}{\em (\cite[Theorem~5.14]{LN97})}
Let $m$ be a positive integer. Let $\chi$ be a nontrivial multiplicative character of $\F_{q}$ and $\chi'$
be the lift of $\chi$ to $\F_{q^m}$, i.e., $\chi'(x)=\chi(\Norm_{\F_{q^m}/\F_q}(x))$
for $x\in \F_{q^m}$, where $\Norm_{\F_{q^m}/\F_q}$ is the norm from $\F_{q^m}$ to $\F_{q}$. Then,
\[
G_{q^m}(\chi')=(-1)^{m-1} G_{q}(\chi)^m.
\]
\end{theorem}
\begin{theorem}
\label{thm:Stickel2}{\em (\cite[Theorem~11.3.5]{BEW97})}
Let $\eta$ be a multiplicative character of order $\ell>1$ of  $\F_{q}$. For  every nontrivial multiplicative character $\chi$ of $\F_{q}$,
\[
G_{q}(\chi)=\frac{G_{q}(\chi^\ell)}{\chi^\ell(\ell)}
\prod_{i=1}^{\ell-1}
\frac{G_{q}(\eta^i)}{G_{q}(\chi\eta^i)}.
\]
\end{theorem}

We will use the following formula later.
\begin{theorem}\label{prop:charaadd}{\em (\cite[Theorem~5.33]{LN97})}
Let $\psi_{\F_q}$ be the canonical additive character of $\F_q$ with $q$ odd, and
let $f(x)=a_2x^2+a_1x+a_0\in \F_q[x]$ with $a_2\not=0$. Then
\[
\sum_{x\in \F_q}\psi_{\F_q}(f(x))=\psi_{\F_q}(a_0-a_1^2(4a_2)^{-1})\eta(a_2)G_q(\eta),
\]
where $\eta$ is the quadratic character of $\F_q$.
\end{theorem}
\section{Basic lifting constructions}
\subsection{Subdifference sets of the Singer difference sets}\label{sec:Quo}
Let $p$ be a prime, $f\geq 1$, $m\geq 2$ be integers and $q=p^f$. Let $L$ be a complete system of coset representatives of $\F_q^\ast$ in $\F_{q^m}^\ast$. We can, and do, choose $L$ in such a way that $\Tr_{q^m/q}(x)=0$ or $1$ for any $x\in L$.
Let
\[
L_0=\{x\in L\,|\,\Tr_{q^m/q}(x)=0\}\mbox{ and } L_1=\{x\in L\,|\,\Tr_{q^m/q}(x)=1\}.
\]
Then,
\begin {equation*}\label{Singer}
H_0=\{\overline{x}\in \F_{q^m}^\ast/\F_q^\ast\,|\,x\in L_0\}
\end{equation*}
represents a hyperplane of the projective space $\PG(m-1,q)$, and it is a so-called {\it Singer difference set} in the cyclic group $\F_{q^m}^\ast/\F_q^\ast$. (Here $\overline{x}=x\F_q^\ast$ represents the
projective point corresponding to the one-dimensional subspace over $\F_q$ spanned by $x$.)

Any nontrivial multiplicative character $\chi$ of exponent $(q^m-1)/(q-1)$ of $\F_{q^m}^\ast$ induces a character of the quotient group $\F_{q^m}^\ast/\F_q^\ast$, which will also denoted by $\chi$. Moreover, every character of $\F_{q^m}^\ast/\F_q^\ast$ arises in this way. By a result given in \cite{Y}, for any nontrivial multiplicative character $\chi$ of exponent $(q^m-1)/(q-1)$ of $\F_{q^m}^\ast$, we have
$$\chi(H_0)=G_{q^m}(\chi)/q.$$

Assume that $N\,|\,(q^m-1)/(q-1)$. Then $$\overline{C_0}:=C_0^{(N,q^m)}/\F_q^\ast\le \F_{q^m}^\ast/\F_q^\ast.$$
Let $S$ be a complete system of coset representatives of $\overline{C_0}$ in $\F_{q^m}^\ast/\F_q^\ast$,  and set $G=\{s\overline{C_0}\,|\,s\in S\}\simeq \F_{q^m}^\ast/C_0^{(N,q^m)}$. For convenience, we will use $\tilde{s}$ to denote
$s\overline{C_0}$.

In the rest of this section, we will assume that $\Cay(\F_{q^m},C_0^{(N,q^m)}) $ is strongly regular, where $N\,|\,(q^m-1)/(q-1)$. Such a strongly regular graph is called {\it cyclotomic}. The following three series of cyclotomic strongly regular graphs were known~\cite{SW}:
\begin{itemize}
\item[(1)] (subfield case) $C_0^{(N,q^m)}=\F_{q^d}^\ast$ where $d\,|\,m$,
\item[(2)] (semi-primitive case) $-1\in \langle p\rangle\le (\Z/N\Z)^\ast$,
\item[(3)] (sporadic case) $\Cay(\F_{q^m},C_0^{(N,q^m)})$ has one of the eleven sets of parameters
given in Table~\ref{Tab1}.
\begin{table}[h]
\caption{Eleven sporadic examples}
\vspace{-0.5cm}
\label{Tab1}
$$
\begin{array}{|c||c|c|c|c|c|c|c|c|c|c|c|}
\hline
N&     11&19 &35      &37 &43   &67       &107&133&163&323&499\\
\hline
q^m&3^5&5^9&3^{12}&7^9&11^7&17^{33}&3^{53}&5^{18}&41^{81}&3^{144}
&5^{249}\\
\hline
\end{array}
$$
\end{table}
\end{itemize}
We mention in passing that Schmidt and White~\cite{SW} conjectured that besides the above three cases, there are no more cyclotomic strongly regular graphs.
\begin{conj}
Let $N\,|\,\frac{q^m-1}{q-1}$ with $N>1$. If $\Cay(\F_{q^m},C_0^{(N,q^m)})$ is
strongly regular, then one of (1), (2), or (3) above holds.
\end{conj}

This conjecture remains open. Some partial results were obtained in \cite{SW}.

Assume that $\Cay(\F_{q^m},C_0^{(N,q^m)}) $ is strongly regular, where $N\,|\,\frac{q^m-1}{q-1}$. Then $|H_0\cap s\overline{C_0}|$, $s\in S$, take exactly two values. It follows that $|H_0\cap s\overline{C_0}|-|H_0\cap \overline{C_0}|=0$ or $\delta$, where $\delta$ is a nonzero integer. For any nontrivial multiplicative character $\chi$ of $\F_{q^m}$ of exponent $N$,
\begin{align*}
\chi(H_0)=\,&\sum_{s\in S}|H_0\cap s\overline{C_0}|\chi(\tilde{s})\\
=\,&\sum_{s\in S}(|H_0\cap s\overline{C_0}|-|H_0\cap \overline{C_0}|)\chi(\tilde{s})\\
=\,&\delta\sum_{s\in S'}\chi(\tilde{s}),
\end{align*}
where
\begin{equation}\label{def:S'}
S'=\{s\in S : |H_0\cap s\overline{C_0}|-|H_0\cap \overline{C_0}|=\delta\}.
\end{equation}
 Thus
\begin{equation}\label{eq:sum}
\sum_{s\in S'}\chi(\tilde{s})=\frac{\chi(H_0)}{\delta}=\frac{G_{q^m}(\chi)}{\delta q}.
\end{equation}
It follows that $\delta$ is a power of $p$. Furthermore, noting that $G_{q^m}(\chi)\overline{G_{q^m}(\chi)}=q^m$, we see that the set $\{\tilde{s}\mid s\in S'\}\subset G$ forms a difference set,
which is called a {\it subdifference set} of $H_0$~\cite{Mc}. Let $\omega$ be a primitive element of $\F_{q^m}$. Then we could choose $S=\{\overline{1}, \overline{\omega}, \ldots , \overline{\omega}^{N-1}\}$, where $\overline{\omega}=\omega\F_q^\ast$. In this way, since $S'$ is a subset of $S$, we define
\begin{equation}\label{eq:subdi}
I=\{0\le i\le N-1\mid \overline{\omega}^i\in S'\}.
\end{equation}
In the rest of the paper, we will also call $I$ a subdifferecne set in $\Z_N$ of the Singer difference set.
\subsection{Two lifting constructions}\label{sec:two}
Let $\gamma$ be a primitive element of $\F_{q^{2m}}$ and set $\omega=\gamma^{q^m+1}$. Then, $\omega$ is a primitive element in $\F_{q^m}$. Let
$C_j^{(N,q^{2m})}=\gamma^j\langle \gamma^N\rangle$, $0\le j\le N-1$.
The following lifting construction was already given in~\cite{MX}. For completeness, we repeat the construction here.
\begin{proposition} \label{theorem:main1}
Assume that $\F_q^\ast\le C_0^{(N,q^m)}\le \F_{q^m}^\ast$ and $\Cay(\F_{q^m},C_0^{(N,q^m)})$ is strongly regular. Let $I$ be the corresponding subdifferecne set defined in \eqref{eq:subdi}.
Let
\begin{equation}\label{ellip}
E=\bigcup_{i\in I}C_i^{(N,q^{2m})}.
\end{equation}
Then $\Cay(\F_{q^{2m}}, E)$ is a strongly regular graph with negative Latin square type parameters $(n^2,r(n+1),-n+r^2+3r,r^2+r)$, where $n=q^m$ and $r=(q^m-1)|I|/N$.
\end{proposition}
\proof
Let $\psi_{\F_{q^{2m}}}$ be the canonical additive character of $\F_{q^{2m}}$ and let  $\chi_N'$ be a multiplicative character of order $N$ of $\F_{q^{2m}}$. We will show that $\psi_{\F_{q^{2m}}}(\gamma^a E)$, $0\leq a\leq N-1$, take exactly two distinct values. By the orthogonality of characters, we compute
\begin{eqnarray*}
S_a=N\cdot \psi_{\F_{q^{2m}}}(\gamma^a E)+|I|=\sum_{j=1}^{N-1}G_{q^{2m}}(\chi_N'^{-j})\sum_{i\in I}\chi_N'^{j}(\gamma^{a+i}).
\end{eqnarray*}
Since $N\,|\,\frac{q^m-1}{q-1}$, there is a multiplicative character $\chi_N$ of $\F_{q^m}$ of order $N$ such that $\chi_N'(\gamma)=\chi_N(\omega)$, i.e., $\chi_N'$ is the lift of $\chi_N$.
By the Davenport-Hasse lifting formula, we have
\[
S_a
=-\sum_{j=1}^{N-1}{\chi_{N}}^{j}(\omega^{a})G_{q^m}(\chi_{N}^{-j})G_{q^m}(\chi_{N}^{-j})
\sum_{i\in I}\chi_{N}^{j}(\omega^{i}).
\]
On the other hand, from the definition of $I$, we have
\begin{equation}\label{eq:defIuse}
\sum_{i\in I}\chi_N^j(\omega^i)=\sum_{s\in S'}\chi_N^j(\tilde{s})=\frac{G_{q^m}(\chi_N^j)}{\delta q}.
\end{equation}
Hence,
\begin{align*}
S_a
=\,&-\frac{1}{\delta q} \sum_{j=1}^{N-1}\chi_{N}^{j}(\omega^{a})G_{q^m}(\chi_{N}^{-j})G_{q^m}(\chi_{N}^{-j})G_{q^m}(\chi_{N}^{j})\nonumber\\
=\,&- \frac{q^{m-1}}{\delta} \sum_{j=1}^{N-1}\chi_{N}^{j}(\omega^{a})G_{q^m}(\chi_{N}^{-j})\\
=\,&-q^m\sum_{j=1}^{N-1}\sum_{i \in I}\chi_N^{-j}(\omega^i)\chi_N^{j}(\omega^a)
=q^m|I|-\begin{cases}
q^m N,& \text{ if }  a\in I, \\
0,& \text{ if } a\not\in I.
\end{cases}
\end{align*}
Thus,  $\psi_{\F_{q^{2m}}}(\gamma^a E)=\frac{S_{a}-|I|}{N}$, $0\leq a\leq N-1$, take exactly two distinct values
$\frac{(q^m-1)|I|}{N}$ and $\frac{(q^m-1)|I|}{N}-q^m$.
Therefore $\Cay(\F_{q^{2m}}, E)$ is strongly regular. The parameters of $\Cay(\F_{q^{2m}}, E)$ can be computed in a straightforward way. We omit the details. \qed
\begin{remark}{\em
If $\Cay(\F_{q^m}, C_0^{(N,q^m)})$ is a cyclotomic strongly regular graph in the subfield case with $N=\frac{q^m-1}{q-1}$, then $C_0^{(N,q^m)}=\F_q^\ast$, $S=\F_{q^m}^\ast/\F_q^\ast$, and $S'=H_0$. In this case, we find that
\[
E=\{x\in \F_{q^{2m}}^\ast\,|\,\Tr_{q^m/q}(x^{q^m+1})=0\},
\] where $\Tr_{q^m/q}(x^{q^m+1})$ is a nondegenerate $\F_q$-valued elliptic quadratic form on $\F_{q^{2m}}$. Therefore it is appropriate to call the lifting construction given in Proposition~\ref{theorem:main1} an {\it elliptic} type lifting construction.}
\end{remark}
We give a new lifting construction, which is  of ``hyperbolic" type.
\begin{proposition} \label{theorem:main2}
Let $\omega$ be a primitive element of $\F_{q^m}$. Assume that $\F_q^\ast\le C_0^{(N,q^m)}\le \F_{q^m}^\ast$ and $\Cay(\F_{q^m},C_0^{(N,q^m)})$ is strongly regular. Let $I$ be the corresponding subdifferecne set defined in \eqref{eq:subdi}. Let
\begin{equation}\label{hyperb}
H=\{(y,y^{-1}x\omega^\ell)\,|\,x \in C_0^{(N,q^m)},y \in \F_{q^m}^\ast,\ell \in I\}\subseteq \F_{q^m}\times \F_{q^m}.
\end{equation}
Then $\Cay(\F_{q^m} \times \F_{q^m}, H)$ is a strongly regular graph with Latin square type parameters  $(n^2,r(n-1),n+r^2-3r,r^2-r)$, where $n=q^m$ and $r=(q^m-1)|I|/N$.
\end{proposition}
\proof
Let $\psi_{\F_{q^m}}$ be the canonical additive character of $\F_{q^{m}}$ and let  $\chi_{q^m-1}$ be a multiplicative character of order $q^m-1$ of $\F_{q^{m}}$.
Each additive character of $\F_{q^m}\times \F_{q^m}$ has the form
\begin{equation}\label{fieldchara22}
\psi_{a,b}((x,y))=\psi_{\F_{q^m}}(ax+by),\; \quad (x,y)\in \F_{q^m}\times \F_{q^m},
\end{equation}
where $(a,b)\in \F_{q^m}\times \F_{q^m}$.
Then, by the definition of $H$, we need to compute the character values:
\[
S_{a,b}:=\sum_{y\in \F_{q^m}^\ast}\sum_{x\in C_0^{(N,q^m)}}\sum_{\ell\in I}\psi_{\F_{q^m}}(ay+bxy^{-1}\omega^\ell), \, \, \quad (0,0)\neq (a,b)\in \F_{q^m}\times \F_{q^m}.
\]
In the case where either one of $a$ or $b$ is zero, it is clear that
$S_{a,b}=-(q^m-1)|I|/N$.

Now, we assume that $a\not=0$ and $b\not=0$.
By the orthogonality of characters, we have
\begin{equation}\label{eq:hypfir1}
S_{a,b}=\frac{1}{(q^m-1)^2}\sum_{j,k=0}^{q^m-2}
\sum_{y\in \F_{q^m}^\ast}\sum_{x\in C_0^{(N,q^m)}}\sum_{\ell\in I}G_{q^m}(\chi_{q^m-1}^{-j})G_{q^m}(\chi_{q^m-1}^{-k})\chi_{q^m-1}^{j}(a)
\chi_{q^m-1}^{k}(bx\omega^\ell)\chi_{q^m-1}^{j-k}(y).
\end{equation}
Since $\sum_{y\in \F_{q^m}^\ast}\chi_{q^m-1}^{j-k}(y)=q^m-1$ or $0$ according as $j\equiv k\,(\mod{q^m-1})$ or $j\not\equiv k\,(\mod{q^m-1})$, continuing from \eqref{eq:hypfir1},
we have
\begin{equation}\label{eq:hypfir2}
S_{a,b}=\frac{1}{q^m-1}\sum_{j=0}^{q^m-2}
\sum_{x\in C_0^{(N,q^m)}}\sum_{\ell\in I}G_{q^m}(\chi_{q^m-1}^{-j})^2\chi_{q^m-1}^{j}(a)
\chi_{q^m-1}^{j}(bx\omega^\ell).
\end{equation}
Let $\chi_{N}:=\chi_{q^m-1}^{\frac{q^m-1}{N}}$.
Since $\sum_{x\in C_0^{(N,q^m)}}\chi_{q^m-1}^{j}(x)=(q^m-1)/N$ or $0$ according as $j\equiv 0\,(\mod{\frac{q^m-1}{N}})$ or $j\not\equiv 0\,(\mod{\frac{q^m-1}{N}})$, continuing from \eqref{eq:hypfir2},
we have
\[
S_{a,b}=\frac{1}{N}\sum_{j=0}^{N-1}
\sum_{\ell\in I}G_{q^m}(\chi_{N}^{-j})^2\chi_{N}^{j}(ab\omega^\ell).
\]
On the other hand, by \eqref{eq:defIuse}, we have $
\sum_{i\in I}\chi_N^j(\omega^i)=\frac{G_{q^m}(\chi_N^j)}{\delta q}$.
Hence, we have
\begin{align*}
S_{a,b}-\frac{|I|}{N}
=\,&\frac{1}{\delta qN} \sum_{j=1}^{N-1}\chi_{N}^{j}(ab)G_{q^m}(\chi_{N}^{-j})G_{q^m}(\chi_{N}^{-j})G_{q^m}(\chi_{N}^{j})\nonumber\\
=\,& \frac{q^{m-1}}{\delta N} \sum_{j=1}^{N-1}\chi_{N}^{j}(\omega^{a})G_{q^m}(\chi_{N}^{-j})\\
=\,&\frac{q^{m}}{N}\sum_{j=1}^{N-1}\sum_{\ell \in I}\chi_N^{-j}(\omega^\ell)\chi_N^{j}(ab)
=-\frac{q^m|I|}{N}+\begin{cases}
q^m,& \text{ if }  \log_{\omega}(ab)\in I\, (\mod{N}), \\
0,& \text{ if } \log_{\omega}(ab)\not\in I\, (\mod{N}).
\end{cases}
\end{align*}
Thus $\psi_{a,b}(H)=S_{a,b}$, $(0,0)\neq (a,b)\in \F_{q^m}\times \F_{q^m}$, take exactly two distinct values $-\frac{(q^m-1)|I|}{N}$ and
$-\frac{(q^m-1)|I|}{N}+q^m$. Therefore $\Cay(\F_{q^m}\times \F_{q^m}, H)$ is strongly regular. The parameters of $\Cay(\F_{q^m}\times \F_{q^m}, H)$ can be computed in a straightforward way. We omit the details. \qed
\begin{remark}{\em
Under the assumptions of Proposition~\ref{theorem:main2}, set
\[
H':=H\cup \{(0,x)\,|\,x \in \F_{q^m}^\ast\}\cup \{(x,0)\,|\,x \in \F_{q^m}^\ast\}.
\]
Then, $\Cay(\F_{q^m}\times \F_{q^m}, H')$ is also strongly regular. Furthermore, if $\Cay(\F_{q^m}, C_0^{(N,q^m)})$ is a cyclotomic strongly regular graph in the subfield case
with $N=\frac{q^m-1}{q-1}$, we have \[
H'=\{(0,0)\neq (x,y)\in \F_{q^m}\times \F_{q^m}\,|\,\Tr_{q^m/q}(xy)=0\},
\] where $\Tr_{q^m/q}(xy)$ is a nondegenerate hyperbolic quadratic form from $\F_{q^m}\times \F_{q^m}$ to $\F_{q}$. Hence, it is appropriate to call the lifting construction in Proposition~\ref{theorem:main2} a {\it hyperbolic} type lifting construction.}
\end{remark}
We now apply Propositions~\ref{theorem:main1} and ~\ref{theorem:main2} to the known cyclotomic strongly regular graphs.
We first apply the two propositions to the semi-primitive examples. In this case,
we have $|I|=1$.

\begin{corollary}\label{cor:semi}
Let $p$ be a prime, $N\ge 2$, $q^m=p^{2js}$, where
$s\ge 2$, $N\,|\,(p^j+1)$, and $j$ is the smallest such positive integer. For $\epsilon\in \{1,-1\}$,  there exists an $(n^2,r(n-\epsilon),\epsilon n+r^2-3\epsilon r,r^2-\epsilon r)$ strongly regular Cayley graph with $n=q^m$ and $r=(q^m-1)/N$.
\end{corollary}
Next we apply Propositions~\ref{theorem:main1} and \ref{theorem:main2}  to the subfield examples. In this case,
we have $N=\frac{q^{m}-1}{q^d-1}$ and $|I|=\frac{q^{m-d}-1}{q^d-1}$, where $d\,|\,m$.
\begin{corollary}\label{cor:singer}
Let $q$ be a prime power and $m\ge 3$ a positive integer.
For $\epsilon\in \{1,-1\}$,  there exists an $(n^2,r(n-\epsilon),\epsilon n+r^2-3\epsilon r,r^2-\epsilon r)$ strongly regular Cayley graph with
$n=q^m$ and $r=q^{m-d}-1$.
\end{corollary}
When $d=1$, the strongly regular graphs obtained in Corollary~\ref{cor:singer} were already known~\cite{Ma}.

Finally, we apply Propositions~\ref{theorem:main1} and \ref{theorem:main2} to the eleven sporadic  examples of cyclotomic strongly regular graphs. In this case,  the values of $k:=|I|$ are
given in \cite[Table II]{SW}.
\begin{corollary}\label{cor-spor}
For $\epsilon\in \{1,-1\}$, there exists an $(n^2,r(n-\epsilon),\epsilon n+r^2-3\epsilon r,r^2-\epsilon r)$ strongly regular Cayley graph with
$n=q^m$ and $r=k(q^m-1)/N$ in each of the following cases:
\begin{eqnarray*}
(q^m,N,k)&=&(3^5,11,5),(5^9,19,9),(3^{12},35,17),(7^9,37,9),(11^7,43,21),(17^{33},67,33)\\
& &(3^{53},107,53),(5^{18},133,33),(41^{81},163,81),(3^{144},323,161),(5^{249},499,249).
\end{eqnarray*}
\end{corollary}

\section{Halving the connection sets $E$ and $H$ and their complements}

In a couple of recent papers \cite{FMX, BLMX}, motivated by existence questions concerning finite geometric objects such as $m$-ovoids and $i$-tight sets,  we used a certain partition of the Singer difference set to construct strongly regular Cayley graphs with special properties which give the desired $m$-ovoids and $i$-tight sets. We now realize that the constructions can be done in a more general setting, namely, we can do the construction by partitioning a subdifference set of the Singer difference set in a certain way. In the case where the cyclotomic strongly regular graph comes from a subfield, the subdifference set of the Singer difference set is actually a Singer difference set; so in this case, we recover the previous constructions. We will also use a certain partition of the complement of a subdifference of the Singer difference sets to construct more strongly regular Cayley graphs.

Assume that $N\geq 2$ is odd, $N\,|\,\frac{q^m-1}{q-1}$, and $\Cay(\F_{q^m},C_0^{(N,q^m)})$ is strongly regular. Let $I$ be the corresponding subdifference set in $\Z_N$ defined in (\ref{eq:subdi}). Let $S_1, S_2$ be a partition of $I$ and
let
$S_i'\equiv 2^{-1}S_i\,(\mod{N})$ and $S_i''\equiv 2^{-1}S_i'\,(\mod{N})$ for
$i=1,2$. Define
\begin{equation}\label{eq:defX1}
X:=2S_1'' \cup (2S_2''+N)\,(\mod{2N}).
\end{equation}
Let $J_1:=\{0,3\}$ and
$J_2:=\{1,2\}$. Define \begin{equation}\label{eq:defI}
Y:=\{Ni+4j \hspace{-0.2cm}\pmod{4N}: (i,j)\in (J_1\times S_1'') \cup (J_2\times S_2'')\}.
\end{equation}
It is clear that $X\equiv 2^{-1}I\,(\mod{N})$ and $Y\equiv I\,(\mod{N})$.

Similarly,  let $T_1, T_2$ be a partition of  $\Z_N\setminus I$ and let
$T_i'\equiv 2^{-1}T_i\,(\mod{N})$ and $T_i''\equiv 2^{-1}T_i'\,(\mod{N})$ for
$i=1,2$.
Define
\begin{equation}\label{eq:defX2}
\widehat{X}:=2T_1'' \cup (2T_2''+N)\,(\mod{2N}).
\end{equation}
Furthermore, define
\begin{equation}\label{eq:defI2}
\widehat{Y}:=\{Ni+4j \hspace{-0.2cm}\pmod{4N}: (i,j)\in (J_1\times T_1'') \cup (J_2\times T_2'')\},
\end{equation}
where $J_1=\{0,3\}$ and $J_2=\{1,2\}$.

\subsection{Decompositions of  $\Cay(\F_{q^{2m}}, E)$ and its complement}\label{sec:minus}
In this subsection, we always assume that $q^m\equiv 3\,(\mod{4})$. We will consider decompositions of  $\Cay(\F_{q^{2m}}, E)$ and its complement, where $E$ is defined in (\ref{ellip}). We define
\begin{equation}\label{eq:defD-}
E_1:=\bigcup_{i \in Y}C_i^{(4N,q^{2m})},
\end{equation}
where $C_i^{(4N,q^{2m})}:=\gamma^i \langle \gamma^{4N}\rangle$, $\gamma$ is a primitive element of $\F_{q^{2m}}$, and $Y$ is defined in \eqref{eq:defI}. Since $Y\equiv I\pmod N$, we see that $E_1$ is a subset of $E$, and $|E_1|=|E|/2$.
The (additive) character values of $E_1$ are given by the following lemma.
\begin{lemma}\label{rem:quad-}
Let $\psi_{\F_{q^{2m}}}$ and $\psi_{\F_{q^m}}$ be the canonical additive characters of $\F_{q^{2m}}$ and $\F_{q^{m}}$, respectively.
For $a\in \Z_{4N}$, define $b\equiv 4^{-1}a\,(\mod{N})$ and $c\equiv 2b\,(\mod{2N})$. Then,
\begin{align}
\psi_{\F_{q^{2m}}}(\gamma^a E_1)=&\,\frac{\rho_p \delta_a q^m}{2G_{q^m}(\eta)}\left(2\psi_{\F_{q^m}}(\omega^c\bigcup_{t\in X}C_t^{(2N,q^m)})-\psi_{\F_{q^m}}
(\omega^c \bigcup_{t\in 2^{-1}I}C_t^{(N,q^m)})\right)\nonumber\\
&\, \hspace{3.0cm}+\frac{(q^m-1)|I|}{2N}-\left\{
\begin{array}{ll}
\frac{q^m}{2}, & \mbox{ if $c \in 2^{-1}I\,(\mod{N})$,}\\
0, & \mbox{ otherwise,}
 \end{array}
\right.\label{eq:DD-}
\end{align}
where $\delta_a=1$ or $-1$ depending on whether {$a\equiv 0,N\,(\mod{4})$ or $a\equiv 2,3N\,(\mod{4})$,} and $\rho_p=1$ or $-1$ depending on
whether $p\equiv 7\,(\mod{8})$ or $p\equiv 3\,(\mod{8})$. Furthermore,
$\eta$ is the quadratic character of $\F_{q^m}$.
\end{lemma}
This lemma is a common generalization of the results in \cite{BLMX} and \cite{Mo}. Its proof is the same as those in \cite{BLMX} and \cite{Mo}. We therefore omit the proof.

Next, we consider a decomposition of the complement of $\Cay(\F_{q^{2m}}, E)$. Let
\begin{equation}\label{eq:defD-2}
E_2:=\bigcup_{i \in \widehat{Y}}C_i^{(4N,q^{2m})},
\end{equation}
where $\widehat{Y}$ is defined in \eqref{eq:defI2}.
The (additive) character values of $E_2$ are given by the following lemma.
\begin{lemma}\label{rem:quad-2}
With the same notation as in Lemma~\ref{rem:quad-},
\begin{align}
\psi_{\F_{q^{2m}}}(\gamma^a E_2)=&\,\frac{\rho_p \delta_a q^m}{2G_{q^m}(\eta)}\left(2\psi_{\F_{q^m}}(\omega^c\bigcup_{t\in \widehat{X}}C_t^{(2N,q^m)})-\psi_{\F_{q^m}}
(\omega^c \bigcup_{t\in \Z_N\setminus 2^{-1}I}C_t^{(N,q^m)})\right)\nonumber\\
&\, \hspace{3.0cm}+\frac{(q^m-1)(N-|I|)}{2}-\left\{
\begin{array}{ll}
0, & \mbox{ if $c \in 2^{-1}I\,(\mod{N})$,}\\
\frac{q^m}{2}, & \mbox{ otherwise.}
 \end{array}
\right.\label{eq:DD-2}
\end{align}
\end{lemma}
\begin{remark}\label{rem:chara}{\em
\begin{itemize}
\item[(i)] If  $X$ defined in \eqref{eq:defX1} satisfies that
\begin{equation}\label{eq:3-chara}
2\psi_{\F_{q^m}}(\omega^c \bigcup_{t\in X}C_{t}^{(2N,q^m)})-
\psi_{\F_{q^m}}(\omega^c \bigcup_{t\in 2^{-1}I}C_{t}^{(N,q^m)})=\left\{
\begin{array}{ll}
\pm G_{q^m}(\eta), & \mbox{ if $c\in 2^{-1}I\, (\mod{N})$,}\\
0, & \mbox{ otherwise, }
 \end{array}
\right.
\end{equation}
substituting \eqref{eq:3-chara} into \eqref{eq:DD-}, we find that the nontrivial additive character values of $E_1$ take two distinct values $\frac{(q^m-1)|I|}{2N}$ and $\frac{(q^m-1)|I|}{2N}-q^m$, implying that $\Cay(\F_{q^{2m}}, E_1)$ is strongly regular.
\item[(ii)] If
$\widehat{X}$ defined in \eqref{eq:defX2} satisfies that
\begin{equation}\label{eq:3-chara2}
2\psi_{\F_{q^m}}(\omega^c \bigcup_{t\in \widehat{X}}C_{t}^{(2N,q^m)})-
\psi_{\F_{q^m}}(\omega^c \bigcup_{t\in \Z_N\setminus 2^{-1}I}C_{t}^{(N,q^m)})=\left\{
\begin{array}{ll}
0, & \mbox{ if $c\in 2^{-1}I\, (\mod{N})$,}\\
\pm G_{q^m}(\eta), & \mbox{ otherwise, }
 \end{array}
\right.
\end{equation}
substituting \eqref{eq:3-chara2} into \eqref{eq:DD-2}, we find that the nontrivial additive character values of $E_2$ take two distinct values $\frac{(q^m-1)(N-|I|)}{2N}$ and $\frac{(q^m-1)(N-|I|)}{2N}-q^m$, implying that $\Cay(\F_{q^{2m}}, E_2)$ is strongly regular.
\end{itemize}}
\end{remark}
\subsection{Decompositions of  $\Cay(\F_{q^{m}}\times \F_{q^{m}}, H)$
and its complement}
In this subsection, we assume that $q^m\equiv 1\,(\mod{4})$, $N$ is an odd divisor of $\frac{q^m -1}{q-1}$,  and $\gcd{(N,\frac{q^m-1}{N})}=1$.
Define
\begin{equation}\label{def:D+}
H_1:=\{(xy,xy^{-1}z\omega^{\ell})\,|\,x\in C_0^{(N,q^m)},y\in C_0^{(\frac{q^m-1}{N},q^m)},z\in C_0^{(4N,q^m)},\ell \in Y\}\subseteq
\F_{q^m}\times \F_{q^m},
\end{equation}
where $\omega$ is a primitive element of $\F_{q^m}$ defined in Subsection~\ref{sec:two} and $Y$ is defined in \eqref{eq:defI}.  In the definition of $H_1$, since $x^2z\omega^\ell \in \bigcup_{\ell \in I}C_\ell^{(N,q^m)}$, we see that $H_1$ is a subset of $H$. Moreover, $|H_1|=|H|/2.$ The (additive) character values of $H_1$ are given in the following lemma.
\begin{lemma}\label{mainconstruction2+}
Let $\psi_{a,b}$ be an additive character of $\F_{q^m}\times \F_{q^m}$ defined in \eqref{fieldchara22} and $\psi_{\F_{q^m}}$ be the canonical additive character of $\F_{q^m}$.
For $(a,b)\in \F_{q^m}\times \F_{q^m}\setminus \{(0,0)\}$ with $ab=0$, it holds that $\psi_{a,b}(H_1)
=-(q^m-1)|I|/2N$. For $(a,b)\in \F_{q^m}\times \F_{q^m}\setminus \{(0,0)\}$ with $ab\not=0$, it holds that
\begin{align}
\psi_{a,b}(H_1)=&\,\frac{\eta(2\omega^c)G_{q^m}(\eta)\delta_{a,b}
}{2}
\left(2\psi_{\F_{q^m}}(\omega^c\bigcup_{t\in X}C_t^{(2N,q^3)})-\psi_{\F_{q^m}}
(\omega^c \bigcup_{t\in 2^{-1}I}C_t^{(N,q^m)})\right)\nonumber\\
&\, \hspace{3.0cm}-\frac{(q^m-1)|I|}{2N}+\left\{
\begin{array}{ll}
\frac{q^m}{2}, & \mbox{ if $c \in 2^{-1}I\,(\mod{N})$,}\\
0, & \mbox{ otherwise,}
 \end{array}
\right.\label{eq:DD+}
\end{align}
where $c$ is defined by  $\omega^c=(ab)^{\frac{N+1}{2}}$ and $\delta_{a,b}=1$ or $-1$ depending on whether {$\log_{\omega}(a^{-1}b)\equiv 0,N\,(\mod{4})$ or $\log_{\omega}(a^{-1}b)\equiv 2,3N\,(\mod{4})$. }
Furthermore,
$\eta$ is the quadratic character of $\F_{q^m}$.
\end{lemma}
This lemma is a generalization of \cite[Theorem~4.1]{FMX}. Since the proof is similar to that of \cite[Theorem~4.2]{FMX}, we omit the proof.

Next, we consider a decomposition of the complement of $\Cay(\F_{q^{m}}\times \F_{q^m}, H)$. Define
\begin{equation}\label{def:D+}
H_2:=\{(xy,xy^{-1}z\omega^{\ell})\,|\,x\in C_0^{(N,q^m)},y\in C_0^{(\frac{q^m-1}{N},q^m)},z\in C_0^{(4N,q^m)},\ell \in \widehat{Y}\}\subseteq
\F_{q^m}\times \F_{q^m},
\end{equation}
where $\widehat{Y}$ is defined in \eqref{eq:defI2}.
The character values of $H_2$ are given in the following lemma.
\begin{lemma}\label{mainconstruction3+}
For $(a,b)\in \F_{q^m}\times \F_{q^m}\setminus \{(0,0)\}$ with $ab=0$, it holds that $\psi_{a,b}(H_2)
=-(q^m-1)(N-|I|)/2N$. For  $(a,b)\in \F_{q^m}\times \F_{q^m}\setminus \{(0,0)\}$ with $ab\not=0$,  it holds that
\begin{align}
\psi_{a,b}(H_2)=&\,\frac{\eta(2\omega^c)G_{q^m}(\eta)\delta_{a,b}
}{2}
\left(2\psi_{\F_{q^m}}(\omega^c\bigcup_{t\in \widehat{X}}C_t^{(2N,q^3)})-\psi_{\F_{q^m}}
(\omega^c \bigcup_{t\in \Z_N\setminus 2^{-1}I}C_t^{(N,q^m)})\right)\nonumber\\
&\, \hspace{3.0cm}-\frac{(q^m-1)(N-|I|)}{2N}+\left\{
\begin{array}{ll}
0, & \mbox{ if $c \in 2^{-1}I\,(\mod{N})$,}\\
\frac{q^m}{2}, & \mbox{ otherwise, }
 \end{array}
\right.\label{eq:DD+2}
\end{align}
where $c$ is defined by $\omega^c=(ab)^{\frac{N+1}{2}}$.
\end{lemma}
\begin{remark}\label{re:cha:hyp}{\em
Similarly to Remark~\ref{rem:chara},
if  the set $X$ defined in \eqref{eq:defX1} satisfies \eqref{eq:3-chara},
the nontrivial additive character values of $H_1$ take two distinct values $-\frac{(q^m-1)|I|}{2N}$ and $-\frac{(q^m-1)|I|}{2N}+q^m$, implying that $\Cay(\F_{q^m}\times \F_{q^m}, H_1)$ is strongly regular.
Also, if
$\widehat{X}$ defined in \eqref{eq:defX2} satisfies
\eqref{eq:3-chara2},
then the nontrivial additive character values of $H_2$ take two distinct values $-\frac{(q^m-1)(N-|I|)}{2N}$ and $-\frac{(q^m-1)(N-|I|)}{2N}+q^m$, implying that $\Cay(\F_{q^m}\times \F_{q^m}, H_2)$ is strongly regular.}
\end{remark}

\section{Partition of subdifference sets and their complements in semi-primitive case}
In this section, we consider a partition of the subdifference sets $I$ in semi-primitive case. We will use the same notation as in Section 4.
We assume that $N$ is odd and $q^{m}=p^{2js}$, where $p$ is a prime,
$s\ge 2$, $N\,|\,(p^j+1)$, and $j$ is the smallest
such positive integer. In this case, $\Cay(\F_{q^{m}},C_0^{(N,q^m)})$ is strongly regular and we have $I=\{0\}$~\cite{SW}.
Furthermore, by Theorem~\ref{thm:semiprim}, the Gauss sums with respect to multiplicative characters of exponent $N$ of $\F_{q^m}$ can be explicitly evaluated as
\begin{equation}\label{eq:Gsemi}
G_{q^m}(\chi_N^i)=(-1)^{s-1}\sqrt{q^m}, \quad 1\le i\le N-1.
\end{equation}
\begin{theorem}\label{thm:semi-}
With the same notation as in Section 4, under the above assumptions, the partition $(S_1,S_2)=(\{0\},\emptyset)$ of $I$ satisfies the
condition~\eqref{eq:3-chara} of Remark~\ref{rem:chara} (i).
\end{theorem}
\proof
By the definition~\eqref{eq:defX1} of $X$, we have $X=\{0\}$.
Write
\[
P_c:=2\psi_{\F_{q^m}}(\omega^c \bigcup_{t\in X}C_t^{(2N,q^m)})-\psi_{\F_{q^m}}(\omega^c \bigcup_{t\in 2^{-1}I}C_t^{(N,q^m)}).
\]
By \eqref{eq:3-chara}, we need to prove that \[
P_c=\left\{
\begin{array}{ll}
\pm G_{q^m}(\eta), & \mbox{ if $c\equiv 0\, (\mod{N})$,}\\
0, & \mbox{ otherwise,}
 \end{array}
\right.
\]
where $\eta$ is the quadratic  character of $\F_{q^m}$.
Let $\chi_{N}$ be a multiplicative character of {order $N$} of $\F_{q^m}$.
By the orthogonality of characters, we have
\begin{align}
P_c=&\,\frac{1}{N}\left(\sum_{i=0}^{N-1}\sum_{j=0,1}\sum_{t\in X}G_{q^m}(\chi_{N}^i\eta^j)\chi_{N}^{-i}\eta^{j}(\omega^{c+t})-\sum_{i=0}^{N-1}\sum_{t\in 2^{-1}I}G_{q^m}(\chi_{N}^i)\chi_{N}^{-i}(\omega^{c+t})\right) \nonumber\\
=&\,\frac{1}{N}\sum_{i=0}^{N-1}\sum_{t\in X}G_{q^m}(\chi_{N}^i\eta)\chi_{N}^{-i}\eta(\omega^{c+t}). \label{eq:hiki}
\end{align}
By the Davenport-Hasse product formula and \eqref{eq:Gsemi}, we have
\[
G_{q^m}(\chi_{N}^i\eta)=\frac{G_{q^m}(\chi_N^{2i})G_{q^m}(\eta)}{G_{q^m}(\chi_{N}^i)}=G_{q^m}(\eta).
\]
On the other hand, by the definition of $X$, we have
$\sum_{t\in X}\chi_{N}^{-i}\eta(\omega^{t})=1$.
Continuing from \eqref{eq:hiki}, we have
\[
P_c=\frac{\eta(\omega^{c})G_{q^m}(\eta)}{N}\sum_{i=0}^{N-1}\chi_{N}^{-i}(\omega^{c})
=
\left\{
\begin{array}{ll}
\eta(\omega^{c})G_{q^m}(\eta), & \mbox{ if $c \equiv 0\,(\mod{N})$,}\\
0, & \mbox{ otherwise.}
 \end{array}
\right. \nonumber
\]
This completes the proof of the theorem.
\qed

\vspace{0.3cm}
Similarly to the theorem above, we have the following.
\begin{theorem}\label{thm:spoc2}
With the notations above,
the partition $(T_1, T_2)=(\Z_N\setminus \{0\},\emptyset)$ of $\Z_N\setminus I$ satisfies the
condition~\eqref{eq:3-chara2}  of Remark~\ref{rem:chara} (ii).
\end{theorem}
Since $q^m=p^{2js}$, we have $q^m\equiv 1\,(\mod{4})$.  We can only apply the lifting construction of hyperbolic type.
By Lemma~\ref{mainconstruction2+}, Remark~\ref{re:cha:hyp} and Theorem~\ref{thm:semi-},  we obtain the following.
\begin{corollary}\label{cor-semi-}
Let $N$ be odd and $q^{m}=p^{2js}$, where $p$ is a prime,
$s\ge 2$, $N\,|\,(p^j+1)$, and $j$ is the smallest
such positive integer. Assume that $\gcd{(N,\frac{q^m-1}{N})}=1$.  Then,
there exists a $(q^{2m},r(q^{m}-1),q^m+r^2-3 r,r^2- r)$ strongly regular Cayley graph, where $r=(q^m-1)/2N$.
\end{corollary}

\vspace{0.3cm}
Similarly to the corollary above, by Lemma~\ref{mainconstruction3+}, Remark~\ref{re:cha:hyp} and Theorem~\ref{thm:spoc2}, we obtain the following corollary.
\begin{corollary}\label{cor-semi-2}
Let $N$ be odd and $q^{m}=p^{2js}$, where $p$ is a prime,
$s\ge 2$, $N\,|\,(p^j+1)$, and $j$ is the smallest
such positive integer. Assume that $\gcd{(N,\frac{q^m-1}{N})}=1$. Then, there exists a $(q^{2m},r(q^{m}-1),q^m+r^2-3 r,r^2- r)$ strongly regular Cayley graph, where  $r=(N-1)(q^m-1)/2N$.
\end{corollary}

\section{Partition of subdifference sets and their complements in sporadic case}
In this section, we consider partitions of the subdifference set $I$ and its complement in the sporadic case.
\begin{theorem}\label{thm:spo}
Assume that $N\geq 2$ is odd, $N\;|\;\frac{q^m-1}{q-1}$, $\Cay(\F_{q^m},C_0^{(N,q^m)})$ is strongly regular and $-2\in \langle p\rangle\,(\mod{N})$, where $p$ is the characteristic of $\F_{q^m}$. Let $I$ be the corresponding subdifference set defined in (\ref{eq:subdi}). Then, the partition $(S_1,S_2)=(I,\emptyset)$ of $I$ satisfies the
condition~\eqref{eq:3-chara} of Remark~\ref{rem:chara} (i).
\end{theorem}
\proof
Let $I'\equiv 4^{-1}I\,(\mod{N})$.
By the definition~\eqref{eq:defX1} of $X$, we have $X\equiv 2I'\,(\mod{2N})$.
Write
\[
P_c:=2\psi_{\F_{q^m}}(\omega^c \bigcup_{t\in X}C_t^{(2N,q^m)})-\psi_{\F_{q^m}}(\omega^c \bigcup_{t\in 2^{-1}I}C_t^{(N,q^m)}).
\]
Let $\chi_{N}$ be a multiplicative character of $\F_{q^m}$ of order $N$ and
$\eta$ be the quadratic  character of $\F_{q^m}$. Similarly to the proof of
Theorem~\ref{thm:semi-}, we have
\begin{align}
P_c=\frac{1}{N}\sum_{i=1}^{N-1}\sum_{t\in X}G_{q^m}(\chi_{N}^i\eta)\chi_{N}^{-i}\eta(\omega^{c+t})+\frac{\eta(\omega^c)G_{q^m}(\eta)|I|}{N}. \label{eq:hiki2}
\end{align}
By the Davenport-Hasse product formula, we have
\[
G_{q^m}(\chi_{N}^i\eta)=\frac{G_{q^m}(\chi_N^{2i})G_{q^m}(\eta)}{G_{q^m}(\chi_{N}^i)}.
\]
On the other hand, by  \eqref{eq:sum},  for $i\not=0$
\[
\sum_{t\in X}\chi_{N}^{-i}\eta(\omega^{t})=\sum_{t\in 2I'}\chi_{N}^{-i}(\omega^{t})=\frac{G_{q^m}(\chi_N^{-2^{-1}i})}{\delta q}.
\]
Substituting these into \eqref{eq:hiki2}, we have
\begin{equation}\label{eq:subdi2}
P_c=\frac{\eta(\omega^{c})G_{q^m}(\eta)}{\delta q N}\sum_{i=1}^{N}\frac{G_{q^m}(\chi_N^{2i})G_{q^m}(\chi_N^{-2^{-1}i})}{G_{q^m}(\chi_{N}^i)}\chi_{N}^{-i}(\omega^{c})+\frac{\eta(\omega^c)G_{q^m}(\eta)|I|}{N}.
\end{equation}
By the assumption that $-2\in \langle p\rangle \,(\mod{N})$, we have
$G_{q^m}(\chi_N^{-2^{-1}i})=G_{q^m}(\chi_N^i)$. Therefore, continuing from
\eqref{eq:subdi2}, we have
\begin{align*}
P_c=&\,\frac{\eta(\omega^{c})G_{q^m}(\eta)}{\delta q N}\sum_{i=1}^{N}G_{q^m}(\chi_N^{2i})\chi_{N}^{-i}(\omega^{c})+\frac{\eta(\omega^c)G_{q^m}(\eta)|I|}{N}\\
=&\,\frac{\eta(\omega^{c})G_{q^m}(\eta)}{N}\left(\sum_{i=1}^{N}\sum_{t\in I}\chi_N^{2i}(\omega^t)\chi_{N}^{-i}(\omega^{c})+|I|\right)\\
=&\,\frac{\eta(\omega^{c})G_{q^m}(\eta)}{N}\sum_{i=0}^{N}\sum_{t\in I}\chi_N^{2i}(\omega^t)\chi_{N}^{-i}(\omega^{c})=
\left\{
\begin{array}{ll}
\eta(\omega^{c})G_{q^m}(\eta), & \mbox{ if $c \in 2I\,(\mod{N})$,}\\
0, & \mbox{ otherwise.}
 \end{array}
\right.
\end{align*}
Since the subdifference set $I$ is invariant under the multiplication by $p$ modulo $N$, by the assumption that $-2^{-1}\in \langle p\rangle \,(\mod{N})$, the condition
$c\in 2I\,(\mod{N})$ is equivalent to that $c\in 2^{-1}I\,(\mod{N})$.
This completes the proof of the theorem.
\qed

\vspace{0.3cm}
Similarly to the theorem above, we have the following.
\begin{theorem}\label{thm:spoc}
Assume that $N\geq 2$ is odd, $N\;|\;\frac{q^m-1}{q-1}$,  $\Cay(\F_{q^m},C_0^{(N,q^m)})$ is strongly regular and $-2\in \langle p\rangle\,(\mod{N})$.
Then the partition $(T_1, T_2)=(\Z_N\setminus I,\emptyset)$ of $\Z_N\setminus I$ satisfies the
condition~\eqref{eq:3-chara2}  of Remark~\ref{rem:chara} (ii).
\end{theorem}
There are ten sporadic examples of cyclotomic strongly regular graphs  satisfying the condition $-2\in \langle p\rangle\,(\mod{N})$. In particular, when
$q^m\equiv 3\,(\mod{4})$, we obtain the following result.
\begin{corollary}\label{cor-spor}
There exists a $(q^{2m},r(q^{m}+1),-q^m+r^2+3 r,r^2+ r)$ strongly regular Cayley graph with $r=k(q^m-1)/2N$  in each of the following cases:
\[
(q^m,N,k)=(3^5,11,5),(11^7,43,21),(3^{53},107,53).
\]
\end{corollary}
\proof
It is clear that $-2\in \langle p\rangle\,(\mod{N})$ in these cases. Then, by applying Lemma~\ref{rem:quad-}, Remark~\ref{rem:chara}~(i)  and  Theorem~\ref{thm:spo} to these examples, the corollary now follows.
\qed

\vspace{0.3cm}
Similarly to the corollary above, by applying Lemma~\ref{rem:quad-2},  Remark~\ref{rem:chara}~(ii) and Theorem~\ref{thm:spoc}  to the three sporadic cyclotomic strongly regular graphs in Corollary~\ref{cor-spor}, we obtain the following.
\begin{corollary}\label{cor-spor2}
There exists a $(q^{2m},r(q^{m}+1),-q^m+r^2+3 r,r^2+ r)$ strongly regular Cayley graph with $r=(N-k)(q^m-1)/2N$  in each of the following cases:
\[
(q^m,N,k)=(3^5,11,5),(11^7,43,21),(3^{53},107,53).
\]
\end{corollary}

\vspace{0.3cm}
In the case where $(q^m,N,k)=(7^9,37,9)$,
the condition that $-2\in \langle p\rangle\,(\mod{N})$ is not satisfied.
We checked by computer that there is no partition of the subdifference set $I$ satisfying the condition~\eqref{eq:3-chara}. On the other hand, we checked that there is a
partition of $\Z_{37}\setminus I$ satisfying the condition~\eqref{eq:3-chara2}: $T_1= 2I$ and
$T_2=\Z_{37}\setminus (I\cup 2I)$.
Hence,
we have the following corollary.
\begin{corollary}\label{cor-spor}
There exists a $(q^{2m},r(q^m+1),-q^m+r^2+3 r,r^2+ r)$ strongly regular Cayley graph with $r=(N-k)(q^m-1)/2N$ in the case where $(q^m,N,k)=(7^9,37,9)$.
\end{corollary}
Next, we consider the case where $q^m\equiv 1\,(\mod{4})$.
\begin{corollary}\label{cor-spor3}
There exists a $(q^{2m},r(q^m-1),q^m+r^2-3r,r^2- r)$ strongly regular Cayley graph with $r=k(q^m-1)/2N$ in each of the following cases:
\begin{align*}
(q^m,N,k)=&\,(3^{12},35,17),(5^9,19,9),(17^{33},67,33), (5^{18},133,33),\\
&\,(41^{81},163,81),(3^{144},323,161),(5^{249},499,249).
\end{align*}
\end{corollary}
\proof
It is clear that  $\gcd{(N,\frac{q^m-1}{N})}=1$ and $-2\in \langle p\rangle\,(\mod{N})$ in these cases.
Then, by applying Lemma~\ref{mainconstruction2+}, Remark~\ref{re:cha:hyp} and  Theorem~\ref{thm:spo} to these examples, the corollary now follows.
\qed

\vspace{0.3cm}
Similarly to the corollary above, by applying Lemma~\ref{mainconstruction3+}, Remark~\ref{re:cha:hyp} and Theorem~\ref{thm:spoc} to these examples, we obtain the following corollary.
\begin{corollary}\label{cor-spor4}
There exists a $(q^{2m},r(q^{m}-1),q^m+r^2-3 r,r^2- r)$ strongly regular Cayley graph with $r=(N-k)(q^m-1)/2N$  in each of the following cases:
\begin{align*}
(q^m,N,k)=&\,(3^{12},35,17),(5^9,19,9),(17^{33},67,33), (5^{18},133,33),\\
&\,(41^{81},163,81),(3^{144},323,161),(5^{249},499,249).
\end{align*}
\end{corollary}
\section{Partitions of subdifference sets and their complements in the subfield case}
In this section, we consider partitions of the subdifference set $I$ and its complement in subfield case. We assume that $m$ is odd and $N=\frac{q^m-1}{q-1}$.
In this case, $\Cay(\F_{q^{m}},C_0^{(N,q^m)})$ is strongly regular and we have
\begin{equation}\label{eq:sub1}
I:=\{i\,(\mod{N}):\Tr_{q^m/q}(w^i)=0\}.
\end{equation}
\subsection{A partition of the Singer difference set $I$ defined in (\ref{eq:sub1}) when $m=3$}
In the case where $m=3$, a partition of the Singer difference set $I$ satisfying the condition~\eqref{eq:3-chara} of Remark~\ref{rem:chara}~(i)  was found in \cite[Theorem~3.7]{FMX}.
Regarding $\F_{q^3}$ as a $3$-dimensional vector space over $\F_q$, we  use $\F_{q^3}$ as the underlying vector space of $\PG(2,q)$. The points of $\PG(2,q)$ are $\langle{\omega^i}\rangle$,
$0\le i\le N-1$, and the lines of $\PG(2,q)$ are
\begin{equation}\label{eqn_Lu}
L_i:=\{\langle{x}\rangle:\,\Tr_{q^3/q}(\omega^i x)=0\},\, \, \, \, 0\le i\le N-1.
\end{equation}
The Singer difference  set $I$ corresponds to the typical line $L_0$.

Consider a nondegenerate quadratic form $f: \F_{q^3}\rightarrow \F_q$ defined by $f(x)=
\tr_{q^3/q}(x^2)$, which
defines a conic $\cQ$ in $\PG(2,q)$ containing $q+1$ points.
Consequently, each line $L$ of $\PG(2,q)$ meets $\cQ$ in $0$, $1$ or $2$ points.
Consider the following subset of $\Z_N$:
\begin{equation}\label{eqn_IQ}
I_\cQ:=\{i\, (\mod{N}):f(\omega^{i})=0\}=\{d_0,d_1,\ldots, d_{q}\},
\end{equation}
where the elements are numbered in any unspecified order.
Thus,  $\cQ=\{\langle{\omega^{d_i}}\rangle:\,0\le i\le q\}$. Furthermore,
by the definition of $f$ and $I$, $I_\cQ\equiv 2^{-1}I\pmod{N}$.

For $d_0\in I_\cQ$, define
\
{\mathcal X}:=\{\omega^{d_i}\Tr_{q^3/q}(\omega^{d_0+d_i}):\,1\le i\le q\}\cup\{2  \omega^{d_0}\}
\
and
\begin{equation}\label{eqn_defX2}
X:=\{\log_{\omega}(x)\, (\mod{2N}):\, x\in {\mathcal X}\}\subset \Z_{2N}.
\end{equation}
Clearly, $|X|=|I_{\mathcal Q}|$ and $X\equiv I_\cQ\pmod{N}$.
If we use any other $d_i$ instead of $d_0$ in the definition of ${\mathcal X}$, then the resulting set $X'$ satisfies that
$X'\equiv X\, (\mod{2N})$ or $X'\equiv X+N\, (\mod{2N})$~\cite[Lemma~3.4]{FMX}.

The set $X$ can be expressed as
\begin{equation}\label{eqn_defX}
X=2S_1''\cup (2S_2''+N)\pmod{2N}
\end{equation}
for some $S_1'',S_2''\subseteq \Z_N$ with $|S_1''|+|S_2''|=q+1$.
Define $S_i'\equiv 2S_i''\pmod{N}$ and $S_i\equiv 2S_i'\pmod{N}$ for
$i=1,2$.
Then,
$S_1'\cup S_2'\equiv I_\cQ\pmod{N}$ and $S_1\cup S_2\equiv I\pmod{N}$, i.e., $X$ induces partitions of $I_\cQ$ and  $I$, respectively.

\begin{theorem}\label{thm:sub6}{\em \cite[Theorem 3.7]{FMX}}
The set $X$ defined in \eqref{eqn_defX2} satisfies the condition \eqref{eq:3-chara}  of Remark~\ref{rem:chara} (i).
\end{theorem}
As corollaries, we have the following.
\begin{corollary}\label{cor:el}
For a prime power $q\equiv 3\,(\mod{4})$, there exists
a $(q^{6},r(q^{3}+1),-q^3+r^2+3 r,r^2+ r)$ strongly regular Cayley graph, where $r=(q^2-1)/2$.
\end{corollary}
\proof
By Lemma~\ref{rem:quad-}, Remark~\ref{rem:chara}~(i)  and  Theorem~\ref{thm:sub6}, the corollary now follows.
\qed

\vspace{0.3cm} The connection set $E_1\subseteq \F_{q^6}$ of the strongly regular Cayley graph $\Cay(\F_{q^6}, E_1)$ obtained in Corollary~\ref{cor:el}
corresponds to a $\frac{(q+1)}{2}$-ovoid in an elliptic quadric ${\mathcal Q}^-(5,q)$. See \cite{BLMX}.
\begin{corollary}\label{cor:hyp}
For a prime power $q\equiv 5,9\,(\mod{12})$, there exists a  $(q^6,r(q^3-1),q^3+r^2-3r,r^2-r)$ strongly regular Cayley graph, where  $r=(q^2-1)/2$.
\end{corollary}
\proof
It is clear that  $\gcd{(N,\frac{q^3-1}{N})}=1$ if $N=q^2+q+1$ and $q\equiv 5,9\,(\mod{12})$.
Then, by Lemma~\ref{mainconstruction2+}, Remark~\ref{rem:chara}~(ii)  and  Theorem~\ref{thm:sub6}, the corollary now follows.
\qed

\vspace{0.3cm}
The connection set {$H_1\subseteq \F_{q^3}\times \F_{q^3}$} of the strongly regular Cayley graph {$\Cay(\F_{q^3}\times \F_{q^3}, H_1)$} obtained in Corollary~\ref{cor:hyp}
corresponds to a $\frac{(q^2-1)}{2}$-tight set in a hyperbolic quadric ${\mathcal Q}^+(5,q)$. See \cite{DDMR,FMX}.

It would be interesting to find a desired partition of $I$ when $m$ is odd and $m>3$. We leave this as an open problem.

\subsection{A partition of the complement of the Singer difference set with odd $m$}
In this section, we consider a partition of the complement of the Singer difference set $I\,(\mod{\frac{q^m-1}{q-1}})$, where  $m>1$ is an arbitrary odd integer. Note that the set $2^{-1}I\,(\mod{\frac{q^m-1}{q-1}})$ corresponds to a nondegenerate parabolic quadric ${\mathcal Q}(m-1,q)$ of $\PG(m-1,q)$.

Let $N=\frac{q^m-1}{q-1}$ and $\omega$ be a primitive element of $\F_{q^m}$, where $q$ is an odd prime power and $m>1$ is an odd integer.
Define
\begin{align*}
A=&\,\{x\in \F_{q^m}^\ast\,|\,\Tr_{q^m/q}(x^{2})= 0\},\\
A_0=&\,\{x\in \F_{q^m}^\ast\,|\,\Tr_{q^m/q}(x^{2})\in C_0^{(2,q)}\},\\
A_1=&\,\{x\in \F_{q^m}^\ast\,|\,\Tr_{q^m/q}(x^{2})\in C_1^{(2,q)}\}.
\end{align*}
Let $a_1\in A$, and define $H_1=\{x\in \F_{q^m}^\ast\mid \Tr_{q^m/q}(a_1x)=0\}$. Note that $A$ represents a nondegenerate parabolic quadric of $\PG(m-1,q)$ and $H_1$ is a tangent hyperplane\footnote{Strictly speaking, we should say that {$H_1\cup\{0\}$ is a hyperplane.}} to $A$ at point $\langle a_1\rangle$. Thus $A\cap H_1$ is a cone of order one with vertex $\langle a_1\rangle$, and $|A\cap H_1|=q^{m-2}-1$. If $m=3$, we stop this process. Otherwise, we continue by choosing $a_2\in A\cap H_1$ such that $a_1,a_2$ are linearly independent over $\F_q$, and define $H_2=\{x\in \F_{q^m}^\ast\mid \Tr_{q^m/q}(a_2x)=0\}$. Then $H_1\cap H_2$ is a hyperplane of $H_1$. Note that $A\cap H_1$ represents a degenerate quadric (a cone of order one) in $H_1$, and $H_1\cap H_2$ contains the vertex $\langle a_1\rangle$, we see that $A\cap H_1\cap H_2$ is a cone of order two (cf. \cite{games}), and $|A\cap H_1\cap H_2|=q^{m-3}-1$. More generally, we define
\begin{align*}
H_{\ell}=&\,\{x\in \F_{q^m}^\ast\,|\,\Tr_{q^m/q}(xa_\ell)= 0\}, \, \quad a_\ell\in A \cap H_{1}\cap \cdots \cap H_{\ell-1},\\
H_{\ell,0}=&\,\{x\in \F_{q^m}^\ast\,|\,\Tr_{q^m/q}(xa_\ell)\in C_0^{(2,q)}\}, \, \quad a_\ell\in A \cap H_{1}\cap \cdots \cap H_{\ell-1},\\
H_{\ell,1}=&\,\{x\in \F_{q^m}^\ast\,|\,\Tr_{q^m/q}(xa_\ell)\in C_1^{(2,q)}\}, \, \quad a_\ell\in A \cap H_{1}\cap \cdots \cap H_{\ell-1},
\end{align*}
where $2\le \ell \le \frac{m-1}{2}$. We can always choose $a_1,\ldots, a_{\frac{m-1}{2}}$ so that they are linearly independent over $\F_q$. The reason is as follows: assume that $a_1,\ldots,a_{\ell-1}$ with $2\le \ell \le \frac{m-1}{2}$ are independent; since $a_1,\ldots,a_{\ell-1} \in A\cap H_{1}\cap \cdots \cap H_{\ell-1}$ and
\begin{equation}\label{eq:numnu}
|A\cap H_{1}\cap \cdots \cap H_{\ell-1}|=q^{m-\ell}-1,
\end{equation}
there are at least $m-\ell$ independent elements in $A\cap H_{1}\cap \cdots \cap H_{\ell-1}$ including $a_1,\ldots,a_{\ell-1}$; hence, we can choose an element $a_{\ell} \in A\cap H_{1}\cap \cdots \cap H_{\ell-1}$ so that $a_1,\ldots,a_\ell$ are independent over $\F_q$ whenever $\ell \le \frac{m-1}{2}$.

Let $b$ be a fixed element of $(H_1\cap \cdots \cap H_{\frac{m-1}{2}})\setminus A$.
Since $H_1\cap \cdots \cap H_{\frac{m-1}{2}}$ and  $A\cap H_1\cap \cdots \cap H_{\frac{m-1}{2}}$ correspond to a $\frac{(m-1)}{2}$-flat and
 a $\frac{(m-3)}{2}$-flat, respectively, in $\PG(m-1,q)$, the set $(H_1\cap \cdots \cap H_{\frac{m-1}{2}})\setminus A$ can be represented as
\
(H_1\cap \cdots \cap H_{\frac{m-1}{2}})\setminus A=\{a_1x_1+\cdots+
a_\frac{m-1}{2} x_\frac{m-1}{2}+by\,|\,x_1,\ldots,x_{\frac{m-1}{2}}\in \F_q, y \in \F_{q}^\ast\}.
\
Let $T_1=(A_0\cap H_{1,0})\cup (A_1\cap H_{1,1})$ and more generally
\[
T_\ell:=(A_0 \cap H_{1} \cap \cdots \cap H_{\ell-1} \cap H_{\ell,0})
\cup (A_1 \cap H_{1} \cap \cdots \cap H_{\ell-1} \cap H_{\ell,1}),
\quad \, 2\le \ell \le \frac{m-1}{2},
\]
and
\[
B:=\{a_1x_1+\cdots+
a_\frac{m-1}{2} x_\frac{m-1}{2}+by\,|\,x_1,\ldots,x_{\frac{m-1}{2}}\in \F_q, y \in C_0^{(2,q)}\}.
\]
Finally,
define \[
D:=\left(\bigcup_{\ell=1}^{(m-1)/2}T_\ell\right)\cup B.
\]
It is clear that
\[
\omega^{N} T_\ell=(A_0 \cap H_{1} \cap \cdots \cap H_{\ell-1} \cap H_{\ell,1})
\cup (A_1 \cap H_{1} \cap \cdots \cap H_{\ell-1} \cap H_{\ell,0})
\]
and \[
\omega^NB=\{a_1x_1+\cdots+
a_\frac{m-1}{2} x_\frac{m-1}{2}+by\,|\,x_1,\ldots,x_{\frac{m-1}{2}}\in \F_q, y \in C_1^{(2,q)}\}.
\]
Hence, $D\cap \omega^{N}D=\emptyset$ and
$D\cup \omega^{N}D=\F_{q^m}^\ast \setminus A$. Thus,
there exists a subset $\widehat{X}\subseteq \Z_{2N}$ such that
$D=\bigcup_{t\in \widehat{X}}C_t^{(2N,q^m)}$ and $\widehat{X}\equiv \Z_N\setminus 2^{-1}I \,(\mod{N})$. The set $\widehat{X}$ induces a partition of  the complement of $2^{-1}I\,(\mod{N})$.
\begin{theorem}\label{thm:coSing}
The set $\widehat{X}$ defined above satisfies the
condition~\eqref{eq:3-chara2}  of Remark~\ref{rem:chara} (ii).
\end{theorem}
To prove this theorem, we need the following lemmas.
\begin{lemma}\label{lem:coSing}
It holds that
\[
\psi_{\F_{q^m}}(\omega^a
T_\ell)=
\left\{
\begin{array}{ll}
{\frac{(-1)^{i+\epsilon \frac{m-1}{2}}q^{\frac{m-1}{2}}(-1+(-1)^{j+\tau} G_q(\eta'))}{2},} & \mbox{ if $\omega^a \in A_i\cap H_1\cap \cdots \cap H_{\ell-1}\cap H_{\ell,j}$, $i,j=0,1,$ }\\
-\frac{q^{m-\ell-1}(q-1)}{2}, & \mbox{ if $\omega^a \in \langle a_1,\ldots,a_\ell\rangle\setminus \langle a_1,\ldots,a_{\ell-1}\rangle$, }\\
\frac{q^{m-\ell-1}(q-1)^2}{2}, & \mbox{ if $\omega^a \in \langle a_1,\ldots,a_{\ell-1}\rangle$, }\\
0, & \mbox{ otherwise, }
 \end{array}
\right.
\]
where $\eta'$ is the quadratic character of $\F_q$ and $\epsilon=0$ or $1$ according as $q\equiv 1\,(\mod{4})$ or $q\equiv 3\,(\mod{4})$.
{Furthermore, $\tau$ is defined by $2\in C_\tau^{(2,q)}$.}
\end{lemma}
The proof of this lemma is complicated. Therefore, we postpone the proof to the Appendix.
\begin{lemma}\label{lem:coSing2}
It holds that
\begin{align*}
\psi_{\F_{q^m}}(\omega^a B)=
\left\{
\begin{array}{ll}
q^{\frac{m-1}{2}}\frac{(-1+G_q(\eta'))}{2}, & \mbox{ if $\Tr_{q^m/q}(\omega^{a}b)\in C_0^{(2,q)}$,
$\omega^a \in (H_1\cap \cdots \cap H_\frac{m-1}{2})\setminus A$, }\\
q^{\frac{m-1}{2}}\frac{(-1-G_q(\eta'))}{2}, & \mbox{ if $\Tr_{q^m/q}(\omega^{a}b)\in C_1^{(2,q)}$,
$\omega^a \in (H_1\cap \cdots \cap H_\frac{m-1}{2})\setminus A$,
}\\
\frac{q^{\frac{m-1}{2}}(q-1)}{2}, & \mbox{ if $\omega^a \in A \cap H_1\cap \cdots \cap H_\frac{m-1}{2}$,}
\\
0, & \mbox{ otherwise.}
 \end{array}
\right.
\end{align*}
\end{lemma}
\proof
We compute the character values of $B$:
\begin{align*}
\psi_{\F_{q^m}}(\omega^a B)=&\,\sum_{x_1,\ldots,x_{\frac{m-1}{2}}\in \F_q}
\sum_{y\in C_0^{(2,q)}}
\psi_{\F_{q^m}}(\omega^a a_1x_1)\cdots \psi_{\F_{q^m}}(\omega^a a_\frac{m-1}{2}x_\frac{m-1}{2})\psi_{\F_{q^m}}(\omega^a b y)\\
=&\,\left(\prod_{i=1}^{\frac{m-1}{2}}\sum_{x_i\in \F_q}
\psi_{\F_{q}}(\Tr_{q^m/q}(\omega^a a_i)x_i)\right)\left(\sum_{y\in C_0^{(2,q)}}\psi_{\F_q}(\Tr_{q^m/q}(\omega^a b) y)\right).
\end{align*}
If  $\Tr_{q^m/q}(\omega^a a_i)\not=0$ for some $i=1,\ldots,\frac{m-1}{2}$, then it is clear that $\psi_{\F_{q^m}}(\omega^a B)=0$. Otherwise, we have
\begin{align*}
\psi_{\F_{q^m}}(\omega^a B)=&\, q^{\frac{m-1}{2}}\sum_{y\in C_0^{(2,q)}}\psi_{\F_q}(\Tr_{q^m/q}(\omega^a b) y)\\
=&\, \left\{
\begin{array}{ll}
\frac{q^{\frac{m-1}{2}}(-1+G_q(\eta'))}{2}, & \mbox{ if $\Tr_{q^m/q}(\omega^{a}b)\in C_0^{(2,q)}$,  }\\
\frac{q^{\frac{m-1}{2}}(-1-G_q(\eta'))}{2}, & \mbox{ if $\Tr_{q^m/q}(\omega^{a}b)\in C_1^{(2,q)}$, }\\
\frac{q^{\frac{m-1}{2}}(q-1)}{2}, & \mbox{ if $\Tr_{q^m/q}(\omega^{a}b)=0$.}
 \end{array}
\right.
\end{align*}
Since $\Tr_{q^m/q}(\omega^a a_1)=\cdots =\Tr_{q^m/q}(\omega^a a_\frac{m-1}{2})=\Tr_{q^m/q}(\omega^a b)=0$ if and only if $\omega^a \in A \cap H_1\cap \cdots \cap H_\frac{m-1}{2}$, the assertion of the lemma follows. \qed

\vspace{0.3cm}
We are now ready to prove Theorem~\ref{thm:coSing}.

{\bf Proof of Theorem~\ref{thm:coSing}:} \,
From Lemmas~\ref{lem:coSing} and \ref{lem:coSing2}, we have
\begin{align*}
\psi_{\F_{q^m}}(\omega^a\bigcup_{t\in \widehat{X}}C_t^{(2N,q^m)})=&\,\psi_{\F_{q^m}}(\omega^a D)=\sum_{i=1}^{\frac{m-1}{2}}\psi_{\F_{q^m}}(\omega^a T_\ell)
+\psi_{\F_{q^m}}(\omega^a B)\\
=&\,
\left\{
\begin{array}{ll}
{\frac{(-1)^{i+\epsilon \frac{m-1}{2}}q^{\frac{m-1}{2}}(-1+(-1)^{j+\tau} G_q(\eta'))}{2},}, & \mbox{ if $\omega^a \in A_i\cap H_1\cap \cdots \cap H_{\ell-1}\cap H_{\ell,j}$ }\\
 & \mbox{ \hspace{1cm}for $\ell=1,\ldots,\frac{m-1}{2}$, $i,j=0,1$, }\\
\frac{q^{\frac{m-1}{2}}(-1+(-1)^i G_q(\eta'))}{2}, & \mbox{ if $\omega^a\in (H_1\cap \cdots \cap H_{\frac{m-1}{2}})\setminus A$ and }\\
 & \mbox{  \hspace{1cm}$\Tr_{q^m/q}(\omega^a b)\in C_i^{(2,q)}$ for $i=0,1,$}\\
0, & \mbox{ if $\omega^a \in A$. }
 \end{array}
\right.
\end{align*}
On the other hand, since $\psi_{\F_{q^m}}(\omega^a \bigcup_{t\in 2^{-1}I}C_t^{(N,q^m)})=\psi_{\F_{q^m}}(\omega^a D)+\psi_{\F_{q^m}}(\omega^{a+N} D)$, we have
\[
\psi_{\F_{q^m}}(\omega^a \bigcup_{t\in 2^{-1}I}C_t^{(N,q^m)})\\
=
\left\{
\begin{array}{ll}
-(-1)^{i+\epsilon \frac{m-1}{2}}q^{\frac{m-1}{2}}, & \mbox{ if $\omega^a \in A_i\setminus (H_1\cap \cdots \cap H_{\frac{m-1}{2}})$, $i=0,1,$}\\
-q^{\frac{m-1}{2}}, & \mbox{ if $\omega^a\in (H_1\cap \cdots \cap H_{\frac{m-1}{2}})\setminus A$.}\\
0, & \mbox{ if $\omega^a \in A$.}
 \end{array}
\right.
\]
Hence, we have
\begin{equation*}
2\psi_{\F_{q^m}}(\omega^a\bigcup_{t\in \widehat{X}}C_t^{(2N,q^m)})-\psi_{\F_{q^m}}(\omega^a \bigcup_{t\in 2^{-1}I}C_t^{(N,q^m)})=
\left\{
\begin{array}{ll}
0, & \mbox{ if $a\in 2^{-1}I\,(\mod{N})$,}\\
\pm q^{\frac{m-1}{2}}G_q(\eta'), & \mbox{ otherwise.}
 \end{array}
\right.
\end{equation*}
Thus,
we conclude that  $\widehat{X}$
satisfies the condition \eqref{eq:3-chara2}  in Remark~\ref{rem:chara} (ii).
\qed

\vspace{0.3cm}
As corollaries, we obtain the following.
\begin{corollary}\label{cor:aff1}
For a prime power $q\equiv 3\,(\mod{4})$ and an odd integer $m>1$, there exists
a $(q^{2m},r(q^{m}+1),-q^m+r^2+3 r,r^2+ r)$ strongly regular Cayley graph with $r=q^{m-1}(q-1)/2$.
\end{corollary}
\proof
By Lemma~\ref{rem:quad-}, Remark~\ref{rem:chara}~(i)  and  Theorem~\ref{thm:coSing}, the corollary now follows.
\qed
\begin{remark} {\em
The strongly regular graph obtained in Corollary~\ref{cor:aff1} has the same parameter as the affine polar graph of elliptic type. Let $\Gamma $
be the strongly regular graph of Corollary~\ref{cor:aff1}  with $q=3$ and $m=3$. We checked by using a computer that  $\Gamma$ is {\bf not} isomorphic to the affine polar graph $\AP^-$ with the same parameters.  In particular, the size of the full automorphism group of $\Gamma$ (resp. $\AP^-$) is $2^2\cdot 3^7 \cdot 7$ (resp. $2^{10}\cdot 3^{12} \cdot 5\cdot 7$). }
\end{remark}

\begin{corollary}\label{cor:aff2}
For a prime power $q\equiv 1\,(\mod{4})$ and an odd integer $m>1$ such that $\gcd{(q-1,\frac{q^m-1}{q-1})}=1$, there exists a  $(q^{2m},r(q^m-1),q^m+r^2-3r,r^2-r)$ strongly regular Cayley graph, where  $r=q^{m-1}(q-1)/2$.
\end{corollary}
\proof
By Lemma~\ref{mainconstruction2+}, Remark~\ref{rem:chara}~(ii)  and  Theorem~\ref{thm:coSing}, the corollary now follows.
\qed
\begin{remark} {\em
The strongly regular graph obtained in Corollary~\ref{cor:aff2} has the same parameters as the affine polar graph of hyperbolic type. Let $\Gamma$
be the strongly regular graph of Corollary~\ref{cor:aff2}  with $q=5$ and $m=3$. We checked by using a computer that  $\Gamma$ is {\bf not} isomorphic to the affine polar graph $\AP^+$ with the same parameters.
In particular, the size of the full automorphism group of $\Gamma$ (resp. $\AP^+$) is $2^3\cdot 5^6 \cdot 31$ (resp. $2^{11} \cdot 3^2\cdot 5^{12}\cdot 13 \cdot 31$). }
\end{remark}

redpajama_set_name: RedPajamaArXiv


text:
Peak sanctuaries
Minoan peak sanctuaries are widespread throughout the island of Crete (Greece). Most scholars agree that peak sanctuaries were used for religious rites. In all peak sanctuaries, human and animal clay figurines have been found. Clay body parts, also called votive body parts, are also found in most peak sanctuaries. These open-air sanctuaries are found high in the mountains of Crete.
1 Eastern and east-central peak sanctuaries
2 Central Crete peak sanctuaries
3 Western Crete peak sanctuaries
4 Other peak sanctuaries
Eastern and east-central peak sanctuaries[edit]
Most peak sanctuaries are found in east and east-central Crete.
Petsofas is the only Minoan site with clay weasel and tortoise figurines
Traostalos
Kalamafki (also Kalamaki)
Ziros Korphi tou Mare
Xykephalo
Vigla (also Viglos)
Zou Prinias
Plagia
Etiani Kephala
Thylakas
Karfi
Central Crete peak sanctuaries[edit]
Iouktas is probably the earliest of the peak sanctuaries.
Tylissos (also Pyrgos Tylissos, not the same site as Pyrgos)
Gonies Philioremos
Western Crete peak sanctuaries[edit]
Vrysinas
Spili Vorizi
Atsipades in the Korakias mountains was fully excavated in the 1980s. Its many hundred clay figurines and other ceramics have been analysed in detail.
Other peak sanctuaries[edit]
This section is for peak sanctuaries mentioned in passing in articles where more research is needed before categorizing them.
Ambelos
Korakomouri
Chryssoulaki, Stella The Traostalos Peak Sanctuary: Aspects of Spatial Organisation Retrieved 19 January 2006
Jones, Donald W. 1999. Peak Sanctuaries and Sacred Caves in Minoan Crete. ISBN 91-7081-153-9
Kyriakidis, Evangelos. 2005. Ritual in the Aegean: The Minoan Peak Sanctuaries. London: Duckworth publishers
Retrieved from "http://en.turkcewiki.org/w/index.php?title=Peak_sanctuaries&oldid=987265335"

redpajama_set_name: RedPajamaCommonCrawl


text:
American Flat Track is America's original extreme sport. At its core, it's a highly competitive, adrenaline-fueled American motorcycle sport featuring customized motorcycles reaching top speeds of up to 140mph, piloted by young, athletic and marketable world-class athletes.
With roots dating back to the first two-wheel speed demons in the 1920s, American Flat Track is widely regarded as the most prestigious and competitive form of dirt track motorcycle racing in the world. A lot has changed over the last 90 years, but the spirit of the sport remains as perhaps the truest, purest test of man and machine. Today, the series is comprised of two classes: AFT Twins and AFT Singles.
On a typical race day, both American Flat Track classes begin the day with practice and qualifying sessions prior to opening ceremonies. From there, it's nothing but exhilarating, full-throttle racing action as the field in each class qualifies for the night's main event through a series of heat and semi races.
In addition to watching the thrilling on-track action, events also feature fan walks where fans are invited into the paddock to meet the stars of the sport and watch the crew chiefs and mechanics at work.
260 GTX is the new spec fuel for 2019 and beyond.
260 GTX


Colorado is not only our home, it's our playground. And here in Crystal Valley, we fully embrace all the ways Colorado allows us to play. That's why we've built in plenty of singletrack trails, acres and acres of open space for exploring, and plenty of other reasons to get you out of the house and into the 300-plus days of sunshine that Colorado has to offer. Explore the Crystal Valley community map, because today's adventure is up to you.
The Recreation Center is bustling year round with community activities, group fitness classes, parties, a fitness center and pool.
From hosting parties to celebrating with neighbors, the Community Room is the hub of the Pinnacle Park Recreation Center.
Who says you need a gym membership, when you have a full fitness center right up the street?
Open from Memorial Day weekend through Labor Day, the community swimming pool is the perfect place to splash, laugh, repeat.
The Tot Splash Pool is a fun and safe way for your little ones to take a little dip.


The reality is, Floyd Mayweather would stand little chance against most MMA fighters in or around his weight category regardless of gender, and next to no chance against Ronday Rousey.
The internet has been ablaze with talk of how a fight between UFC super star Ronday Rousey and boxing pay per view king Floyd Mayweather would pan out if the two were to meet in a no holds barred fight. Both athletes are at the top of their respective games, and the age old question as to which fighting style is the best makes for an intriguing match up.
The fact that Ronday Rousey is a woman also makes a proposed fight more interesting given the supposed limitations of female fighters - a stereotype Rousey has obliterated with her dominant performances in the octagon.
Height and reach plays a factor in fights, and even a solitary inch can make a big difference. Taller fighters with longer reach can reach their opponents first, and given Mayweather's advantages in speed over Rousey, the extra inches in both height and reach would play a significant factor in a fight. Mayweather's weight would also play a big factor given his added muscle would translate directly in to power.
Undoubtedly Mayweather is the faster athlete. Not only is Mayweather fast compared to Rousey, he is fast compared to the fastest boxers of his generation. His incredible quickness would present huge difficulties for Rousey, who would undoubtedly eat some razor sharp punches as she tried to work her way in.
I swear on my life her back muscles were the most solidest back muscles I'd ever felt in my life. I thought if this lady was to get a hold of me, she would throw me on my head in literally one second flat. So I would not like to engage in a clinch fight with Ronda.
Floyd Mayweather is strong when it comes to boxing, but Rousey's strength is on a totally different level. If Mayweather got anywhere near Rousey, he'd find himself flat on the floor in a matter of mili-seconds. There simply is not comparison.
While Ronda Rousey is a very smart fighter with elite grappling intelligence, there is no avoiding the fact that Floyd Mayweather might be the most intelligent fighter in the history of boxing. Mayweather has literal psychic abilities in the ring and is able to not only adapt on a moment by moment basis to his opponent, but think several rounds ahead.
However, boxing intelligence and MMA intelligence are two totally separate things, and while Mayweather is a savant in his own sport, he would be a complete beginner in kicking, clinching and grappling - arts Rousey is far, far more educated in. You can't diminish Mayweather's fighting intelligence in any way, but in an MMA match he is no genius.
There is no comparison here. Floyd Mayweather is one of the greatest boxers in history. Ronda Rousey has improved her boxing skills dramatically, but if she chose to duke it out with Mayweather from distance, she'd get beaten very badly.
While Ronday Rousey doesn't kick much, she is well versed in Muay Thai so knows what she is doing. As far as anyone knows, Mayweather has never thrown a kick in his life, and doesn't spend any of his week training with elite kick boxers.
Mayweather is fantastic at clinching in boxing - he ties opponents up, blocks punches with his elbows, and always finds a way to beat people on the inside. However, this does not compare to Rousey's clinch skills which are far more varied and would allow her to take Mayweather to the ground instantaneously.
There is no discussion here. Ronda Rousey has finished almost all of her fights via submission, throws men around in practice, and has won a gold medal at the World Junior Judo Championships, won the bronze medal at the Junior World Championships, won a silver medal at the 2007 World Judo Championships, and won a bronze medal at the 2008 Olympics. Rousey is an elite grappler and would maul Mayweather in a matter of seconds on the floor.
When it comes to intangibles like heart, determination and grit, it is hard to separate the two. Both Floyd Mayweather and Ronda Rousey are elite fighters and world champions in their respective sports. We haven't seen Rousey under too much pressure as of yet, while Mayweather has fought and beaten the very best of his generation. So from what we know Mayweather is the more 'proven' of the two.
While the criteria above helps look at each fighter's individual strengths and weaknesses, much of it means very little when it comes to assessing how a fight between the two of them would pan out.
While Mayweather would use his elite speed, footwork and reflexes to stay away from Rousey in an MMA fight, keeping someone as determined, strong and relentless as Rousey would be nearly impossible unless the ring was the size of a football field. Unless Mayweather could knock Rousey out quickly, he would find himself manhandled to the floor and promptly submitted via arm bar (or any other technique Rousey decided to use). Boxers have historically failed incredibly badly when attempting to make the transition to MMA, and Mayweather would be no different.
Men and women almost never fight against each other in competition, and for good reason. The genetic differences in muscle mass, bone structure and strength put women at a significant disadvantage against male athletes, and for safety reasons competition between them it is almost always a very bad idea. However, Ronda Rousey is not a normal woman - she is a freak athlete who defies gender and can compete with elite men, particularly when it comes to grappling.


Lexus is – again – coming with great news to the Paris Motor Show, and I'm extremely happy to stand in front of you today!
If we look back over the last couple of years, Lexus underwent a tremendous change.
We have fully rejuvenated our line-up and added exciting new models such as NX and RC, as well as a growing range of performance cars including the RC F and the GS F.
Today, we offer the broadest line-up of premium hybrid vehicles in the industry - no less than 10 models globally! We sell more premium hybrid vehicles than any other manufacturer, by far!
And our region leads: no less than 98% of Lexus sales in Western Europe are hybrids.
Our design has become much bolder, our cars more emotional, and this, while retaining Lexus' ultimate craftsmanship and quality credentials.
And we are very pleased to see that our efforts are met with enthusiast consumer response on the global market. In Europe for instance, the home of the premium car market, with probably the most demanding customers, Lexus has a clear lead in many customer satisfaction surveys.
This also results in a sustained sales momentum, with double-digit year-on-year sales growth. We are well on our way to exceed our target of 70 thousand sales in Europe this year, a new record for our brand.
But Lexus transformation is only starting to unfold and there's much more to come in the near future to establish Lexus as a truly global luxury lifestyle brand.
The LC, our new flagship coupe, which we revealed earlier this year, is the symbol of this journey. LC is pure emotion, both from design and driving perspectives, supported by innovative technologies.
With LC, we introduce our next generation performance hybrid system: the Lexus Multi Stage Hybrid System. This new powertrain brings more power, greater acceleration higher efficiency, and last but certainly not least, more fun to drive.
LC is coming on the market next year, but this summer we already took it to an extensive tour in Europe, from the Nürburgring to Goodwood, from Madrid to Amsterdam, to connect with customers and car enthusiasts. We wanted to hear their first-hand feedback, and their response was fantastic!
Our next product launch is just around the corner and I am very happy to present it today, for the first time in Europe: the new Lexus IS sports sedan.
The IS is one of Lexus' most successful models ever. Since the launch of the first generation, back in 1999, we have delivered more than one million units to consumers worldwide.
The new IS features revised exterior styling, an even more refined cabin and enhanced equipment such as a multimedia screen up to 10.3 inches.
Also, we have revised suspensions and steering for an even better dynamic experience, and the new IS now includes our benchmark package of active safety technologies, the Lexus Safety System +.
You can expect the new IS in Lexus showrooms by January next year.
Now… time to look deeper into the future! And I'm referring to the vehicle hidden behind this screen, which you will discover in a few moments.
This vehicle is all about anticipation but it is also rooted in Lexus' unparalleled expertise in the SUV market.
In the 90's Lexus pioneered the premium SUV world with the RX and the LX, for the first time in the industry combining off-road capabilities with premium luxury. Then in 2004, we launched the RX hybrid, the first premium hybrid SUV on the market.
In 2014 we further expanded our line-up with NX demonstrating that 'Utility' can be combined with a unique, striking design personality. And NX is already our best-seller in Europe!
From the outset our vision for SUVs was based on luxury and quality, then expanded to innovative hybrid technology and striking design.
Now we are ready for the next chapter - even bolder, even more emotional and unique on the market.
I am very proud to introduce today, in world premiere, our vision of the premium urban crossover of the future. Ladies and gentlemen, here is the Lexus UX Concept!
The dynamic compact SUV packaging is designed to appeal to progressive, urban customers who live in an ever-connected environment.
The UX Concept delivers an immersive driver experience with futuristic 3D holographic displays and other innovative electronic features. The seat design is inspired by Lexus' Kinetic Seat Concept … a ground breaking technology that stimulates the drivers' senses by involving the whole body in the driving experience.
This is what we call a design statement! It's clearly a Lexus, you can recognize all our signature elements: the spindle grille, the L-shaped lights, the striking lines… but in an entirely new execution, and one that will turn heads.
And by the way, the UX Concept was conceived here in France, in ED2, our European Design Centre. Our design philosophy has Japanese roots, but it is also truly global in its expression.
As highlighted by this new concept, Lexus always seeks to anticipate. We are passionate about cars … about design, quality, driving emotions, and not to forget … customer experience. In the coming months and years, we will work even harder to transmit our passion to consumers worldwide.


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