homogeneity and isotropy on sufficiently large scales is an assumption which is fundamental to our current understanding of the universe .this assumption which is popularly known as the cosmological principle implies that the properties of the universe is same for all observers irrespective of their locations and the directions at which they are looking at .the assumption has also an aesthetic appeal of its own as it indirectly asserts us that the same physical laws must apply throughout the universe given it is homogeneous and isotropic to start with .one can not prove the cosmological principle in a strictly mathematical sense but fortunately one can test it against the presently available cosmological observations .the existence of the cosmic microwave background radiation and its near uniform temperature over the entire sky provides by far the best conclusive evidence in favour of isotropy .other important supporting evidences come from the isotropy in angular distributions of radio sources , isotropy in the x - ray background and the isotropy in the distribution of neutral hydrogen as shown by lyman- transmitted flux .but mere existence of isotropy around us does not automatically guarantees homogeneity of the universe .spatial homogeneity can be asserted from isotropy only when the later is confirmed around each point with the hypothesis that the matter distribution is a smooth function of position .we see structures in the present universe starting from planets , stars and galaxies to groups , clusters and superclusters spanning a wide range of length scales .the present universe looks highly inhomogeneous on small scales and it is important to test if such inhomogeneities continue to persist on large scales .large scale inhomogeneities have several important implications for cosmology and could pose a serious challenge to the standard cosmological framework .inhomogeneities , through the backreaction mechanism can provide an alternate explanation of a global cosmic acceleration without requiring any additional dark energy component .a large void can also mimic an apparent acceleration of expansion . fortunately such models can be constrained with observations such as sne , cmb , bao and measurements of angular diameter distance and hubble parameter .however if our universe is inhomogeneous on large scales , the currently available inhomogeneous cosmological models suggest that the effects of inhomogeneities on observed quantities are non - negligible and more precise observations will not be properly analyzed unless inhomogeneities are taken into account .galaxy surveys map the three dimensional distribution of galaxies in the universe. if our universe is homogeneous then the statistical properties of galaxy distributions in a finite volume should be independent of the location of that volume in the universe .one simple tool to characterize the statistical properties of galaxy distributions is the two point correlation function which measures the excess probability of finding a pair of points separated by a distance compared with a random poisson process .the two point correlation function on small scales , is well described by a power law of the form , with correlation length and slope  . vanishes at scales which is consistent with large scale homogeneity .however the problem with correlation function analysis is that it assumes a mean density on the scale of survey which is not a defined quantity below the scale of homogeneity .most of the statistical tests of homogeneity carried out so far are based on the count in spheres method where the number counts within a sphere of radius and its scaling with is used to test the transition scale to homogeneity . if the scaling exponent approaches a value on some scale it marks the transition to homogeneity indicating that the distribution is homogeneous on and above that scale .fractal analysis uses the scaling of different moments of to characterize the scale of homogeneity .various studies carried out with these methods claim to have found a transition to homogeneity on sufficiently large scales  whereas there are studies which claim the absence of any such transition out to the scale of the survey .the present generation galaxy surveys ( sdss , ; 2dfgrs , ) have now mapped the distribution of millions of galaxies across billions of light years providing an unique opportunity to test the assumption of homogeneity on large scales . the sloan digital sky survey ( sdss )has reached its final stage ( data release 12 ) now encompassing more than one - third of the entire celestial sphere .this provides us with galaxy distributions in enormous volumes of the universe for which the statistical properties of galaxy distributions can be analyzed and homogeneity of the universe can be tested with greater confidence . a method based on shannon entropy for characterizing inhomogeneities and applied the method on some monte carlo simulations of inhomogeneous distributions and n - body simulations which show that the proposed method has great potential for testing the large scale homogeneity in galaxy redshift surveys .  in the present work we employ the method proposed by to analyze a set of volume limited samples from the sloan digital sky survey data release twelve ( sdss dr12 ) .we quantify the inhomogeneities in the galaxy distribution using shannon entropy and test if there is a scale of transition to homogeneity .we also analyze a set of mock galaxy samples from a semi analytic galaxy catalogue derived from the millennium run simulation to compare theoretical predictions with observations .  a brief outline of the paper follows .we briefly describe our method in section 2 , describe the monte carlo simulation , sdss and millennium data in section 3 and present the results and conclusions in section 4 .we have used a cosmological model with , and throughout . propose a method based on the shannon entropy to study inhomogeneities in a 3d distribution of points .shannon entropy is originally proposed by claude shannon to quantify the information content in strings of text .it gives a measure of the amount of information required to describe a random variable .the shannon entropy for a discrete random variable with outcomes is a measure of uncertainty denoted by defined as ,    where is the probability distribution of the random variable .given a set of points distributed in 3d we consider each of the points as center and determine the number of other points within a sphere of radius as , + where is the heaviside step function and and are the radius vector of and points respectively . to avoid any edge effects we discard all the centers which lie within a distance from the survey boundary .the number of available centers will decrease with increasing for any finite volume sample .we define a separate random variable for each radius which has possible outcomes each given by , with the constraint .the shannon entropy associated with the random variable can be written as , where the base of the logarithm is arbitrary and we choose it to be . will have the same value for all the centers when is same for all of them .this is an ideal situation when the spheres of radius around each of the centres contain exactly the same number of points .this maximizes the shannon entropy to for radius .we define the relative shannon entropy as the ratio of the entropy of a random variable to the maximum possible entropy associated with it .the relative shannon entropy at any quantifies the degree of uncertainty in the knowledge of the random variable .the distribution of become completely uniform when is reached .  in the present scenario the random variables are not mutually independent as the measurements around different centers at each radius are carried out on the same finite volume sample and multiple spheres around the centers can overlap .the random variables may have extra correlations if the points are clustered or distributed in an intrinsically inhomogeneous way .these extra correlations increase the mutual information stored in the random variables .the decrease of information in with increasing is evident irrespective of a homogeneous / inhomogeneous distribution but it would diminish differently depending on the nature and the degree of inhomogeneity present in the distribution . when inhomogeneities are present , the relative shannon entropy would show a departure from the values .larger departure clearly indicates greater degree of inhomogeneity .when the shannon entropy attains its maximum value corresponding to that radius this ratio levels up with .this indicates absence of inhomogeneity beyond that scale and may be considered as the scale of transition to homogeneity .but this should be taken with some caution as overlap between spheres around different centres forcibly brings down the inhomogeneities by increasing the relative shannon entropy at each length scales .overlapping become progressively more important at larger radii .the number of available centres at a radius gradually decreases with increasing and the centres preferentially migrate towards the centre of the volume analyzed .the available centres are finally confined to a smaller region when the largest spheres are considered .this confinement bias may finally force the relative shannon entropy to at some radius when the spheres completely overlap with each other .this corresponds to a situation where the spheres around different centres show a near uniform number count mimicking a real transition to homogeneity .the degree of overlap as well as the confinement bias would depend on the shape of the volume and nature of the distribution to some extent .a large volume ensures spheres upto a larger length scales and hence may help us to detect the scale of homogeneity if the transition occurs before the scale where the confinement bias completely dominates the statistics . forthe present analysis one may also consider non - overlapping spheres of different radii .but the statistics becomes too noisy due to very small number of independent spheres at progressively larger radii which consequently prohibits us to address the issue of homogeneity on large scales .the observed inhomogeneities are the combined outcome of gravitational clustering , poisson noise and any intrinsic inhomogeneity present in the distribution .inhomogeneity due to poisson noise rapidly decreases with .the inhomogeneity due to clustering would also decrease with albeit differently depending on the strength and nature of clustering .overlap causes a further suppression in the inhomogeneities at all scales . a complete discussion on the various sources of inhomogeneity and the factors affecting the value of the relative shannon entropy can be found in .+   +  +application of the method discussed in section 2 . involves analysis of galaxy distributions in finite regions of the universe .construction of volume limited samples from galaxy surveys over different redshift and magnitude ranges would result in samples having different volumes and numbers of galaxies .one may also prefer to analyze volume limited galaxy samples having different geometries . keeping these possibilities in mindwe would like to first investigate how the parameters like number density , volume and geometry affect the relative shannon entropy of a 3d spatial distribution of a point process .  to study the possible systematic effects we generate different sets of monte carlo realizations of the homogeneous poisson point process .we consider the following cases : ( i ) number density is allowed to vary keeping the volume and geometry fixed , ( ii ) volume is allowed to vary keeping the number density and geometry fixed and ( iii ) geometry is allowed to vary keeping both the number density and volume fixed .we simulate 10 monte carlo realizations for each density , volume and geometry considered . , and summarize the properties of the monte carlo simulations generated to test the effects of number density , volume and geometry respectively .  [ cols="^,^,^,^,^,^ " , ]   semi analytic models provide a very powerful tool to study galaxy formation and evolution .galaxy formation and evolution involve many physical processes such as gas cooling , star formation , supernovae feedback , metal enrichment , merging and morphological evolution .the semi analytic models parametrise the physics involved in terms of simple models following the dark matter merger trees over time .the models provide the statistical predictions of galaxy properties at some epoch and the precision of these predictions are directly related to the accuracy of the input physics . in the present analysis we use a semi - analytic galaxy catalogue generated by from the millennium run simulation who updated the previously available galaxy formation models with improved versions .the spectra and magnitude of the model galaxies were computed using population synthesis models of .we map the galaxies to redshift space using their peculiar velocities and then identify regions which have the same geometry as our sdss samples .we then apply the same magnitude cuts as those used for the actual data and extracted the same number of galaxies as in our sdss samples . for each of the volume limited samples used in our analysis we generate three mock galaxy samples from the semi analytic galaxy catalogue .in the top left we show the variations of the relative shannon entropy with distance for some homogeneous poisson point processes generated inside a cubic box with different number densities ( ) . the relative shannon entropy versus curves show a characteristic behaviour in all cases . at the smallest radiusthe relative shannon entropy starts from a value which is very close to .it suddenly drops at a particular length scale and then starts increasing again as the length scale increases .the behaviour of shannon entropy at the smallest radius and the presence of a characteristic dip in it on a certain length scale can be understood as follows .any distribution having a finite number density will look nearly uniform below the scale of mean inter - particle separation corresponding to that distribution . as this characteristic length scaleis surpassed the irregularities or any deviations from homogeneity can be clearly seen .the dips in the set of curves in all panels of are located at the mean inter - particle separations of the respective samples . while testing the effects of number density we have kept unaltered all the other factors such as volume and the geometry of the samples . as the density of the sample drops the characteristic dip in the relative shannon entropy shifts towards larger length scales ( top left panel of ) simply resulting from an increase in the mean inter - particle separation .the differences in the amplitudes of the relative shannon entropy for poisson samples with different densities result from the discreteness noise or shot noise due to poisson fluctuations .as the density of the sample decreases the poisson noise shoots up at each length scale causing a decrease in the relative shannon entropy and increase in the degree of inhomogeneity at the corresponding scales .the points are uncorrelated in case of a poisson point process .the void probability function for a homogeneous poisson distribution is where is the intensity of the poisson process and is the volume .it is expected that such a distribution would become homogeneous on a scale , which is a measure of the average size of voids in the distribution .this is closely related to the mean inter - particle separation and one would expect a poisson distribution to become homogeneous above this length scale .but one can see in the top left panel of that the relative shannon entropy of all the poisson point processes continue to show a departure from upto some distance even after this length scale .this is due to the poisson noise which rapidly decreases with increasing .further the spheres around different centres overlap with each other which reduces the inhomogeneity increasing the relative shannon entropy .overlapping also becomes progressively important at larger length scales in presence of confinement bias .the poisson noise and overlapping affect the relative shannon entropy in opposite manner in which the former enhances whereas the later reduces inhomogeneities at each length scales .it may be noted that the poisson noise dominates at smaller scales whereas overlapping dominates at larger scales .this eventually allows the overlapping to wipe out the inhomogeneities introduced by the poisson noise .it can be seen in the top left panel of that the relative shannon entropy eventually level up with for all the poisson point processes but at different length scales .the lowest density sample having the largest poisson noise at each length scale become homogeneous on the largest length scale among the samples analyzed .we have used here an extreme variation in density for this test , the highest density samples being times denser than the lowest density samples ( see ) .we also test the affects of the size of volume occupied by the distribution .we drastically vary the volume of the samples keeping their number density and geometry unchanged ( see ) .the results are shown in the top right panel of .we can see that the relative shannon entropy are largely unaffected by the change in volume .it shows a very small increase in relative shannon entropy nearly on all scales , the changes being noticeable only when the volume is increased by a factor of .this small change in shannon entropy can be attributed to the cosmic variance as the number of centers available at a particular radius increases with the volume .the largest length scale that can be probed with these samples are different as they cover different volumes ( ) .it is interesting to note that all the samples become homogeneous at nearly the same length scales despite the fact that the samples with different volumes probe upto different length scales and have different number of centers at each radii .although these samples cover different volumes they have exactly the same number density ensuring the same contribution from poisson noise at each length scales .this clearly demonstrates that the poisson noise or discreteness noise governed by the number density of the samples plays an important role in deciding the degree of inhomogeneity and transition to homogeneity .finally we test the effects of geometry of the volumes on the relative shannon entropy .we use a survey like region , a cubic box and a spherical region all having the same number density and volume ( see ) .the results are shown in the lower middle panel of .we see that the degree of inhomogeneity and the transition scale to homogeneity both are unaffected by the change in geometry of the samples . the results of these tests clearly indicate that the relative shannon entropy of a distribution is most sensitive to the number density of the distribution and nearly insensitive to the volume and the geometry of the samples .  in we show the relative shannon entropy at different length scales for the first sdss samples described in together with that from their mock counterparts from the millennium simulation and poisson point processes .these four different sdss volume limited samples are constructed by keeping the brighter magnitude limit fixed at while gradually shifting the fainter magnitude limit from to in steps of .this ensures minimal differences in the clustering properties of the resulting samples while allowing us to probe different length scales due to their different volume coverages .the mock samples drawn from the millennium simulation and poisson processes have identical number density , volume and geometry as their respective actual sdss counterparts .we see a characteristic dip in the relative shannon entropy curve for all the samples at a length scale which corresponds to their mean inter - particle separations ( ) . further in all the panels of we see that the galaxy samples from both the sdss and the millennium have a higher degree of inhomogeneity as compared to their mock poisson samples .these differences indicate that the galaxy samples must have extra sources of inhomogeneity other than the poisson noise resulting purely from the discrete nature of the distributions .it is well known that galaxy distributions exhibit clustering which can be quantified by their correlation functions .clustering acts as an extra source of inhomogeneity for the galaxy samples .besides there could be an intrinsic inhomogeneity built in the distribution .the inhomogeneity from clustering and intrinsic inhomogeneous nature if any combine together with that from poisson noise to produce the total inhomogeneities in the distributions .irrespective of their nature inhomogeneities are expected to decrease with increasing length scales in presence of overlapping and confinement bias .consequently these together makes it difficult to distinguish a real transition to homogeneity from an induced one .the transition can be confirmed easily if it takes place at a length scale where confinement bias does not dominate the statistics .for example in the mock poisson samples corresponding to four sdss samples show a transition to homogeneity in the range .although the transitions are affected by poisson noise on small scales still we can make a decision on the existence of a genuine transition to homogeneity .galaxy distributions are far from a random poisson distribution andif a scale of transition to homogeneity exist there it would possibly occur on sufficiently large scales where the confinement bias may overpower the inhomogeneities .interestingly one can reduce the impact of confinement bias by simply choosing a volume which are less symmetric than a sphere or a cube . although we have mentioned earlier that the geometry of the analyzed volume does not affect the relative shannon entropy as shown in, it could actually affect the statistics if the distribution is inhomogeneous nearly upto the largest length scale that can be probed with that volume . a volume like a rectangular slabwhose two dimensions are significantly larger than the third one would make it sure that the available centres are spread across nearly the entire volume and would therefore never allows a situation where the spheres completely overlap with each other even at the largest radius .thus if the number counts around different centres differ even slightly the relative shannon entropy will show a deviation from .the geometry of our sdss samples are not exactly slab like but definitely has a lesser symmetry than a spherical or cubic volume .this may enable us to detect the signature of any inhomogeneities present at the largest length scales probed by these samples .although overlapping would be present at each scale , it may not be able to erase the inhomogeneities completely even at the largest scale due to the lesser impact of the confinement bias .any residual inhomogeneity detected at the largest length scales are less likely to borne out of pure poisson noise . at the largest length scalethe numbers of spheres could be drastically small so we decided to consider only upto the length scales where this number reduces to . inwe show how the number of centers vary with length scales for all the six sdss samples analyzed here .we can see that different samples probe upto different length scales and the scale where the number of available centres reduces to for sample 1 , sample 2 , sample 3 and sample 4 are , , and respectively .in the top left panel of the relative shannon entropy for the sdss sample 1 clearly shows a deviation from unity at signaling some residual inhomogeneities on that scale .the results of the mock poisson samples shown in indicate that for the galaxy samples having number densities comparable to our sdss samples the contribution from poisson noise can be safely neglected beyond .further sample 1 being the densest of the sdss samples analyzed here would be least affected by the poisson noise at all length scales .this indicates that the galaxy distribution is not homogeneous on .the results from sdss sample 2 , sample 3 and sample 4 are shown in the top right , bottom left and bottom right panel of respectively .sample 2 , sample 3 and sample 4 are , and times less denser than sample 1 ( ) and are expected to have a greater contribution from poisson noise at all length scales .interestingly the relative shannon entropy for sample 1 , sample 2 , sample 3 and sample 4 are nearly same at . as sample 2 ,sample 3 and sample 4 probe progressively larger length scales we see that the relative shannon entropy increases further with increasing length scales. it may appear as if the reduction in inhomogeneity with increasing length scales as revealed by progressively larger samples could result from a greater degree of overlap between the spheres .but the samples probing larger scales have larger volumes and for each of them we restrict our attention only upto a scale where the number of centers reduces to .so the degree of overlap would not vary across the samples given we limit our attention upto this specific length scale .all the four panels of the show that the relative shannon entropy change very slowly on large scales indicating that the degree of overlap do not change rapidly for our sample geometries possibly due to a less aggressive confinement bias .it may be noted further that the values of relative shannon entropy for sample 2 , sample 3 and sample 4 are nearly same at and they are also same at for sample 3 and sample 4 .these together clearly demonstrate the decreasing importance of poisson noise with increasing length scales and indicate that any residual inhomogeneities at the largest length scale could be safely attributed to the presence of genuine inhomogeneity in the distribution at the corresponding length scale .we identify the transition scale to homogeneity corresponding to which the value of lies within  .this criteria is satisfied at for the sample 4 .we also test if this result holds for galaxy samples having sizes bigger than sample 4 and therefore probing larger length scales .we construct another two volume limited samples namely sample 5 and sample 6 ( see ) which are larger than sample 4 .we could not prepare the mock samples for sample 5 and sample 6 from the millennium simulation due to the limitations arising out of the size of the simulation volume .however we analyze sample 5 and sample 6 and find that regardless of their volumes and number densities the distributions become homogeneous at .this can be clearly seen in where we find that the value of the relative shannon entropy of sample 4 , sample 5 and sample 6 agree within at conforming to homogeneity .importantly there are centers available at for both sample 5 and sample 6 which reduces to at and respectively .the observed offset between the curves below the scale of may indicate the presence of a sample size dependent finite size effect . to test thiswe applied the same cut in the upper redshift limit of sample 5 and sample 6 as applied for sample 4 ( ) . this ensures same volume and geometry for these samples as they have the same lower redshift limit and sky coverage .we reanalyzed the resulting samples and find that the offset between the curves persists with very little changes in the differences and all three curves finally merges at ( ) .the offset between the curves primarily originate from the differences in the number density and clustering in the galaxy distributions .sample 5 and sample 6 have brighter magnitude limits and lower number densities than sample 4 .poisson noise induced inhomogeneities resulting from different number densities plays an important role on small scales and become less important as the radius increases ( ) . differences in inhomogeneities due to the differences in clustering may persist upto a large length scale .the fact that all three curves from sample 4 , sample 5 and sample 6 reconcile at indicates that the differences in their inhomogeneities cease to exist at this length scale .the length scale where all three curves merge reduces to in this case ( ) .this results from the decrease in the available number of centres at all length scales when the sample 5 and sample 6 are trimmed .it may be noted in and that the number of available centers reduces from to at and for sample 5 and sample 6 respectively when they are trimmed to match with the size of sample 4 .it may be interesting to note here that an analysis of filamentarity in the luminous red galaxy ( lrg ) distribution found that the filaments are statistically significant upto a length scale of  .some recent studies of the distribution of nearby rich galaxy clusters and superclusters in the sdss find the evidence of shell like structures on . in different panels of we see that the mock galaxy samples from the millennium simulation also does not become homogeneous at least up to a length scale of and become homogeneous at .we note that all the sdss samples exhibit a greater degree of inhomogeneity as compared to their mock counterparts from the millennium simulation .nevertheless we can not deny that the complex issues of overlapping and confinement bias cast some uncertainty in the interpretations which can be resolved with yet larger samples with a reasonable number density .  in futurewe plan to carry out analysis with the luminous red galaxy distributions ( lrg ) and the baryon oscillation spectroscopic survey ( boss ) from sdss which cover enormous volumes providing opportunities to probe much larger scales .noticeably this would also enable us to perform the analysis with non - overlapping independent spheres even at sufficiently larger length scales .analysis with non - overlapping spheres would have a remarkable advantage over the overlapping spheres as in the later the inhomogeneities are suppressed at each length scales making it difficult to detect them particularly at larger scales .this would help us to discern any inhomogeneities present even at the largest scales with nearly equal ease and settle the issue of cosmic homogeneity with an unprecedented confidence .funding for sdss - iii has been provided by the alfred p. sloan foundation , the participating institutions , the national science foundation , and the u.s .department of energy office of science .the sdss - iii web site is http://www.sdss3.org/.  sdss - iii is managed by the astrophysical research consortium for the participating institutions of the sdss - iii collaboration including the university of arizona , the brazilian participation group , brookhaven national laboratory , carnegie mellon university , university of florida , the french participation group , the german participation group , harvard university , the instituto de astrofisica de canarias , the michigan state / notre dame / jina participation group , johns hopkins university , lawrence berkeley national laboratory , max planck institute for astrophysics , max planck institute for extraterrestrial physics , new mexico state university , new york university , ohio state university , pennsylvania state university , university of portsmouth , princeton university , the spanish participation group , university of tokyo , university of utah , vanderbilt university , university of virginia , university of washington , and yale university .

Abstract: we analyze a set of volume limited samples from sdss dr12 to quantify the degree of inhomogeneity at different length scales using shannon entropy . we find that the galaxy distributions exhibit a higher degree of inhomogeneity as compared to a poisson point process at all length scales . our analysis indicates that signatures of inhomogeneities in the galaxy distributions persist at least upto a length scale of . the galaxy distributions appear to be homogeneous on a scale of and beyond . analyzing a set of mock galaxy samples from a semi analytic galaxy catalogue from the millennium simulation we find a scale of transition to homogeneity at .   [ firstpage ]  methods : numerical - galaxies : statistics - cosmology : theory - large scale structure of the universe .


cloud radio - access network ( c - ran ) is a promising architecture to meet the demand for higher data rates in next generation wireless networks . in these systems , the base - stations act as relays and are connected via high - speed backhaul links to a cloud network .encoding and decoding operations happen centrally in the cloud .the study of fundamental information theoretic limits and optimal coding techniques for such systems is a fertile area of research .  motivated by c - ran , we study a model where the source is connected to relay terminals using orthogonal links with a fixed capacity .the relays transmit to the destination over a wireless multiple - access channel .such a setup is known as the diamond - relay network  .we study this model in the presence of an eavesdropper who can eavesdrop the orthogonal links from the source to the relays as well as the wireless transmission from the relays .we require that a message be transmitted reliably to the legitimate receiver , while keeping it confidential from the eavesdropper .we adopt the information - theoretic notion of confidentiality , widely used in the literature on the wiretap channel  .we thus refer to our setup as the _ wiretapped diamond - relay channel_.  while the secrecy capacity of both the scalar gaussian wiretap channel and its multi - antenna extension can be achieved using gaussian codebooks , the optimal schemes in multiuser channels are considerably more intricate .very recently , xie and ulukus  studied the secure degrees of freedom ( d.o.f . ) for one - hop gaussian multiuser channels such as the multiple - access channel and the interference channel .it turns out that interference alignment plays a central role in achieving the optimal secure d.o.f . while coding schemes based on gaussian codebooks can only achieve zero secure d.o.f ., the schemes presented in  involve transmitting a combination of information and jamming signals at each transmitter and judiciously precoding them such that the noise symbols align at the legitimate receiver(s ) , yet mask information symbols at the eavesdropper .this approach has been extended for the case with no eavesdropper s channel state information ( csi ) at the legitimate parties in  .we note that a combination of jamming and interference alignment is also required in the confidential mimo broadcast channel with delayed channel state information  and the compound mimo wiretap channel  .the diamond - relay channel that we consider is a two - hop network where the first hop consists of orthogonal links from the source to the relays , while the second hop is a multiple - access channel from the relays to the destination .such a network is considerably different from one - hop networks as the channel inputs from the relays need not be mutually independent .the source can transmit common / independent message , common noise , or any combination of those to the relays . in  , for the diamond - relay channel where an eavesdropper can wiretap_ only _ the multiple - access portion of the channel , it is shown that transmitting common noise over the source - relay links facilitates more efficient jamming for both the case with full csi and the case with no eavesdropper s csi .for example , a key constituent scheme proposed in for the case with no eavesdropper s csi is computation for jamming ( coj ) where the source transmits a function of information and common noise symbols to two relays in order that the common noise symbols introduced to jam information symbols at the eavesdropper can be canceled at the destination through suitable precoding .when there are more than two relays , this scheme operates in a time - sharing basis in a way that only two source - relay links are utilized in each sub - scheme . in the present paper , the optimal secure d.o.f . of the wiretapped diamond - relay channelis established for the case with no eavesdropper s csi . in our wiretapped diamond - relay channel ,the eavesdropper not only observes the relay transmissions through another multiple - access channel , but also observes a certain number of source - relay links .the legitimate terminals do not know which subset of source - relay links are revealed to the eavesdropper , except for the total number of such links . for achievability , we note that it is not straightforward to extend the proposed schemes in because their sub - schemes require _ asymmetric _ link d.o.f .and hence the amount of wiretapped information varies over time according to the unknown location of wiretapped links .thus , we first develop symmetric version of the proposed schemes in  and then combine them with a secure network coding ( snc )  - like scheme to account for non - secure source - relay links .while the conventional snc has been developed for fully wired networks , we incorporate snc by utilizing the nature of wireless networks in a way that the additional randomness introduced in the source - relay links due to snc is canceled at the destination by beamforming over the wireless multiple - access channel . to that end, we judiciously choose the generator matrix for snc . from a technical point of view, the secrecy analysis involves accounting for the observations from the source - relay links as well as the multiple - access wiretap channel and is considerably more involved .furthermore , the secure d.o.f . is shown to be the same even when the knowledge of the compromised links is available .indeed our converse is established via this genie - aided approach .the rest of this paper is organized as follows . in section[ sec : model ] , we formally state our model of wiretapped diamond - relay channel . the main result on the secure d.o.f . is presented in section [ sec : main ] . for the achievability part, our proposed schemes are described at a high - level in section [ sec : ach ] and are rigorously stated in appendix [ appendix : achievability ] .the converse part is proved in section [ sec : conv ] .we conclude this paper in section [ sec : conclusion ] .the following notation is used throughout the paper . fortwo integers and ,  ] , denotes the vector .this notation is naturally extended for vectors and random variables . for a sequence of constants indexed by time, for positive integer denotes the vector ) ] , ] , if , and 1 if .consider a diamond - relay channel that consists of an orthogonal broadcast channel from a source to relays and a gaussian multiple - access channel from the relays to a destination . in the broadcast part ,the source is connected to each relay through an orthogonal link of capacity . in the multiple - access part ,the channel output at time is given as in which is the channel input at relay , s are channel fading coefficients , and is additive gaussian noise with zero mean and unit variance . the transmit power constraint at relay given as , where denotes the number of channel uses .in this paper , we consider a scenario illustrated in fig .[ fig : model ] , where an eavesdropper wiretaps both the broadcast part and the multiple - access part of the diamond - relay channel .in the broadcast part , the eavesdropper can wiretap source - relay links .let denote the number of secure source - relay links .we assume that the location of wiretapped links is unknown to the source , relays , and destination . in the multiple - access part, the eavesdropper observes at time given as where s are channel fading coefficients and is additive gaussian noise with zero mean and unit variance .+  we assume a fast fading scenario where s and s are drawn in an i.i.d .fashion over time according to an arbitrary real - valued joint density function satisfying that ( i ) all joint and conditional density functions are bounded and ( ii ) there exists a positive finite number such that for all ], ] , we use  ] . ] a stochastic encoder at the source that ( randomly ) maps  ] , a stochastic encoder at time  ] that ( randomly ) maps to , and a decoding function at the destination that ( randomly ) maps to  ] such that .the secrecy capacity is the supremum of all achievable secrecy rates .we also consider the case where the location of wiretapped links is known to all parties .let denote the index set of relays whose links from the source are wiretapped . in this case ,an achievable secrecy rate is defined by requiring to be satisfied for the known  .  in this paper , we are interested in asymptotic behavior of the secrecy capacity when tends to infinity .we say a d.o.f .tuple is achievable if a secrecy rate such that is achievable when . a secure d.o.f . is the maximum such that is achievable .according to the context , denotes .our model guarantees secrecy from a certain level of relay collusion , i.e. , from collusion of any set of up to relays when the location of wiretapped links is unknown and from collusion of any set of relays that have wiretapped links when it is known .our achievability results hold for constant channel gains as well as slow - fading channels .furthermore , our achievability results can be generalized for complex channel fading coefficients by applying ( * ? ? ?* lemma 7 ) in our analysis of interference alignment .in this section , we state the main result of this paper .let us first present the secure d.o.f . of the wiretapped diamond - relay channelwhen , which was established in ( * ? ?* theorem 6 ) .[ thm : same ] the secure d.o.f . of the wiretapped diamond - relay channel when , i.e. , the eavesdropper does not wiretap the broadcast part , is equal to   for the achievability part of theorem [ thm : same ] , two constituent schemes were proposed in .one scheme is a time - shared version of the blind cooperative jamming ( bcj ) scheme , where in each sub - scheme , a single relay operates as a source and the other relays operate as helpers according to the bcj scheme .the other scheme is a time - shared version of the coj scheme , where in each sub - scheme , the source and a pair of relays operate according to the coj scheme ( * ? ? ?* scheme 5 ) while the other relays remain idle . for the converse part of theorem [ thm : same ] ,a technique was introduced in that captures the trade - off between the message rate and the amount of individual randomness injected at each relay .the following theorem establishes the secure d.o.f .of the wiretapped diamond - relay channel for the general case .[ thm : less ] the secure d.o.f . of the wiretapped diamond - relay channel is equal to for both the cases with and without the knowledge of location of wiretapped links .theorem [ thm : less ] indicates that the secure d.o.f .is the same for the cases with and without the knowledge of location of wiretapped links . according to theorem [ thm : less ] , the secure d.o.f .is at most when the number of secure links is one .when the number of secure links is one and its location is known , a natural strategy would be to send the message over the secure link and send nothing over the wiretapped links .then , the multiple - access part becomes the wiretap channel with helpers whose secure d.o.f . is shown to be in .when , we can interpret theorem [ thm : less ] in a way that the secure d.o.f .is decreased as if the link d.o.f . was decreased by a factor of . because only out of links are secure, it is intuitive that the information that can be securely sent over the broadcast part is decreased by a factor of . in fig .[ fig : thm2 ] , the secure d.o.f . of the wiretapped diamond - relay channelis illustrated when and .   and ., title="fig:",width=453 ] +  the achievability part of theorem [ thm : less ] when the location of wiretapped links is known can be easily proved by generalizing the proposed schemes in . however ,when the location of wiretapped links is unknown , it is not straightforward to extend the proposed schemes in because their sub - schemes require _ asymmetric _ link d.o.f . andhence the amount of wiretapped information depends on the unknown location of wiretapped links . to resolve this issue, we first propose _ simultaneous _ bcj ( s - bcj ) and _ simultaneous _ coj ( s - coj ) schemes for the case with the knowledge of location of wiretapped links .then we incorporate snc on top of those schemes for the case without the knowledge of location of wiretapped links .our proposed schemes are described at a high - level in section [ sec : ach ] and rigorously stated in appendix [ appendix : achievability ] .the converse part of theorem [ thm : less ] is proved in section [ sec : conv ] .to prove the achievability part of theorem [ thm : less ] , it suffices to show the achievability of the following corner points : for and for ., can be shown to be achievable by time - sharing between and . for , can be shown to be achievable by time - sharing among , , and . ]  in the following , we first propose s - bcj scheme achieving and s - coj scheme achieving for the special case of . then, we generalize them for the general case of with the knowledge of location of wiretapped links .subsequently , we incorporate snc on top of the s - bcj and s - coj schemes for the case of without the knowledge of location of wiretapped links . to provide main intuition behind our schemes ,let us give a high - level description of our schemes in this section .a detailed description with rigorous analysis is relegated to appendix [ appendix : achievability ] .  in fig . [fig : sbcj ] and fig .[ fig : scoj ] , our proposed schemes are illustrated for , where rectangles labeled with , , represent message , noise , and fictitious message symbols , respectively , and a rectangle labeled with represents a linear combination of fictitious message symbols .a column of symbols with fat side lines implies that those symbols are aligned and occupy the d.o.f . of a single symbol .the source represents the message of d.o.f . as a vector , j\in [ 1:m-1]) ] to relay , which requires .then , relay sends together with independent noise symbols ) ] , s for  ] to relay , which requires .then , the relays send what they have received in a way that ( i ) each of s is beam - formed in the null space of the destination s channel and ( ii ) s can be distinguished by the destination . since s are not aligned and occupy a total of 1 d.o.f at the eavesdropper , the message can be shown to be secure .   in this case , we send nothing over the wiretapped links . without loss of generality ,let us assume that the first links are secure .the source represents the message of d.o.f . as a vector , j\in [ 1:m-1]) ] to relay  ] , which requires .then , relay  ] each of d.o.f . in a way that ( i ) for each  ] are aligned at the destination and ( ii ) s can be distinguished by the destination . since sare not aligned and occupy a total of 1 d.o.f at the eavesdropper , the message can be shown to be secure .the source represents the message of d.o.f .1 as a vector of independent message symbols and generates independent noise symbols , where each and has a d.o.f .the source sends _ n}) ] and sends nothing to relay  ] of independent message symbols each of d.o.f . . to mask each symbol ,we introduce a vector ) ] of linear combinations of fictitious message symbols by computing for some integer matrix .now , the source sends ) ] and sends ) ] , which requires .then , relay  ] each of d.o.f . in a way that ( i ) for each  ] are aligned at the destination , ( ii ) each of s is beam - formed in the null space of the destination s channel , and ( iii ) s can be distinguished by the destination . by judiciously choosing the generator matrix , it can be shown that the information leakage is zero ( in the d.o.f .sense ) .   as in the s - coj scheme when the last links are wiretapped , the source represents the message of d.o.f .1 as a vector of independent message symbols and generates independent noise symbols , where each and has a d.o.f . . then , for a vector ) ] of linear combinations of fictitious message symbols is generated by computing for some integer matrix .now , the source sends _ n}+l_{2k}) ] and sends to relay  ] .furthermore , we assume that in addition to is available at the destination , which only possibly increases the secure d.o.f .hence , and are conditioned in every entropy and mutual information terms in this section , but are omitted for notational convenience . in the following , s for are used to denote positive constants that do not depend on and .first , we obtain } ) + nc_1\\ & \overset{(b)}{\leq } i(w;y_1^n , s^n_{[n+1:m ] } ) - i(w;y_2^n , s^n_{[n+1:m ] } ) + nc_2 \label{eqn : public_fano}\\ & \leq i(w;y_1^n|y_2^n , s^n_{[n+1:m ] } ) + nc_2 \label{eqn : one_dof}\\ & \leq h(y_1^n|y_2^n , s^n_{[n+1:m]})+ nc_2\label{eqn : public_1}\\ & = h(y_1^n , y_2^n|s^n_{[n+1:m ] } ) - h(y_2^n|s^n_{[n+1:m ] } ) + nc_2 \\ & \leq h(x_{[1:m]}^n , y_1^n , y_2^n|s^n_{[n+1:m ] } ) - h(y_2^n|s^n_{[n+1:m ] } ) + nc_2\\ & = h(x_{[1:m]}^n|s^n_{[n+1:m]})+ h(y_1^n , y_2^n|x_{[1:m]}^n , s^n_{[n+1:m ] } ) - h(y_2^n|s^n_{[n+1:m ] } ) + nc_2\\ & \overset{(c)}{= } h(x_{[1:m]}^n|s^n_{[n+1:m ] } ) - h(y_2^n|s^n_{[n+1:m ] } ) + nc_2\label{eqn : same}\\ & \leq h(x_{[1:m]}^n , s^n_{[1:n]}|s^n_{[n+1:m ] ] } )  - h(y_2^n|s^n_{[n+1:m ] } ) + nc_2\\ & \leq h(s^n_{[1:n]})+ h(x_{[1:m]}^n|s^n_{[1:m ] } ) - h(y_2^n|s^n_{[n+1:m ] } ) + nc_2\\ & \leq nnc + \sum_{i=1}^mh(x_i^n|s^n_i)- h(y_2^n|s^n_{[n+1:m]})+nc_2 .\label{eqn : cov12}\end{aligned}\ ] ] where is due to the fano s inequality , is from the secrecy constraint , and is because the deterministic model is assumed .next , to bound for  ] , is from ( * ? ? ?* lemma 2 ) , and is due to the markov chain and are conditioned in every entropy and mutual information terms in this section . ] .now , by combining and , we have } ) -h(y_2^n|s^n_{[n+1:m]})+nc_{5 } \\ & = nnc+(m-1)h(y_1^n|s^n_{[n+1:m]})+h(y_1^n|s^n_{[n+1:m]})-h(y_2^n|s^n_{[n+1:m]})+nc_{5}. \label{eqn : diff_pre}\end{aligned}\ ] ] furthermore , })-h(y_2^n|s^n_{[n+1:m]}) ] denotes the set of all possible distributions of codewords satisfying the power constraint and is from ( * ? ? ?* section 6 ) .  by substituting to ,we obtain . on the other hand , from, we have .therefore ,   for the special case of , to derive a tighter bound , we start from : }^n|s^n_{[2:m ] } ) - h(y_2^n|s^n_{[2:m ] } ) + nc_2\\ & \leq h(x_{1}^n|s^n_{[2:m]})+\sum_{i=2}^m h(x_{i}^n|s^n_{i } ) - h(y_2^n|s^n_{[2:m ] } ) + nc_2 .\label{eqn : cov_1}\end{aligned}\ ] ]  now , by combining and , we have })+ ( m-1)h(y_1^n|s^n_{[2:m]})-h(y_2^n|s^n_{[2:m]})+nc_6 \\ & = h(x_{1}^n|s^n_{[2:m]})+ ( m-2)h(y_1^n|s^n_{[2:m ] } ) + h(y_1^n|s^n_{[2:m]})-h(y_2^n|s^n_{[2:m]})+nc_6 \\ & \overset{(a)}{\leq }  h(x_{1}^n|s^n_{[2:m]})+ ( m-2)h(y_1^n|s^n_{[2:m ] } ) + n\cdot o(\log p)+nc_6\\  & \overset{(b)}\leq  h(x_{1}^n|x^n_{[2:m]})+ ( m-2)h(y_1^n|s^n_{[2:m]})+n\cdot o(\log p)+nc_6 , \label{eqn : cov_3 } \end{aligned}\ ] ] where is from similar steps used to obtain and is due to the markov chain },\mathbf{h}^n , \mathbf{g}^n)-x^n_{[2:m]} ] , we have }^n\big)\\ & =  h\big(\big\ { \lfloor h_1(t ) x_1(t)\rfloor\big\}_{t=1}^n |x_{[2:m]}^n\big)\\ & = h ( x_1^n , \big\ { \lfloor h_1(t ) x_1(t)\rfloor\big\}_{t=1}^n  |x_{[2:m]}^n ) -h(x_1^n|\big\ { \lfloor h_1(t ) x_1(t)\rfloor\big\}_{t=1}^n , x_{[2:m]}^n)\\ & = h ( x_1^n  | x_{[2:m]}^n ) -h(x_1^n|\big\ { \lfloor h_1(t ) x_1(t)\rfloor\big\}_{t=1}^n , x_{[2:m]}^n)\\ & \geq h ( x_1^n  | x_{[2:m]}^n)-\sum_{t=1}^nh(x_1(t)|\lfloor h_1(t ) x_1(t)\rfloor)\\ & \overset{(a)}{\geq } h ( x_1^n| x_{[2:m]}^n)-nc_7 , \label{eqn : cov_4}\end{aligned}\ ] ] where is from ( * ? ? ?* lemma 2 ) . by combining and, we obtain })+n\cdot o(\log p)+nc_8\\ & \leq ( m-1)h(y_1^n)+n\cdot o(\log p)+nc_8 . \ ] ] hence , .therefore , for , we have which concludes the proof .we note that the proof technique used for the special case of can be extended for the general case of , but the resultant bound is loose when .in this paper , we established the optimal secure d.o.f . of the wiretapped diamond - relay channel where an external eavesdropper not only wiretaps the relay transmissions in the multiple - access part , but also observes a certain number of source - relay links in the broadcast part .the secure d.o.f . was shown to be the same for the cases with and without the knowledge of location of wiretapped links .for the former case , we proposed two constituent jamming schemes .the first scheme is the s - bcj scheme where the noise symbols are aligned at the destination and the second scheme is the s - coj scheme where the noise symbols are beam - formed in the null space of the destination s channel . for the latter case where the location of wiretapped links is unknown , we combined a snc - like scheme with s - bcj and s - coj in a way that ( i ) the fictitious message symbols of snc mask all the message and artificial noise symbols transmitted over the source - relay links and ( ii ) the fictitious message symbols can be canceled at the destination via beamforming at the relays .we believe that our results on the wiretapped diamond - relay channel are an important step towards understanding the secrecy capacity of general multi - terminal networks , which remains a largely open problem till date .let us first state two theorems that give us achievable secrecy rates for the wiretapped diamond - relay channel .these theorems are obtained by assuming symbol - wise relay operations and then considering the wiretapped diamond - relay channel as the wiretap channel .these theorems are proved in the end of this appendix .[ thm : sec_rate ] for the wiretapped diamond - relay channel with the knowledge of location of wiretapped links , a secrecy rate is achievable if for some }|v)\prod_{k\in[1:m]}p(x_k|s_k,\mathbf{h}) ] for  ] such that , for some }|v)\prod_{k\in[1:m]}p(x_k|s_k,\mathbf{h}) ] for  ] : , j\in [ 1:m-1])\\ s_k&=\begin{cases } ( v_{k , j } : j\in [ 1:m-1 ] ) , & k\in [ 1:n]\\ \emptyset , & k\in [ n+1:m ] \end{cases}\\ x_k&=\begin{cases } \sum_{j\in [ 1:m-1 ] } \mu_{k , j } v_{k , j } + \sum_{j\in [ 1:n ] } \frac{\nu_j}{h_k}u_{k , j } , & k\in [ 1:n]\\ \sum_{j\in [ 1:n ] } \frac{\nu_j}{h_k}u_{k , j } , & k\in [ n+1:m ] \end{cases}\end{aligned}\ ] ] where s and s are independently generated according to  ] .we note that and \leq p ] are satisfied if where .then , the channel outputs are given as + z_1 \label{eqn : y1_less}\\ y_2&= \sum_{k=1}^n\sum_{j=1}^{m-1 } g_k\mu_{k , j } v_{k , j } + \sum_{k=1}^m \sum_{j=1}^n \frac{g_k\nu_j}{h_k}u_{k , j } + z_2.\end{aligned}\ ] ] because , theorem [ thm : sec_rate ] says that the following secrecy rate is achievable : to derive a lower bound on the rhs of , let us derive a lower and an upper bounds on the first and the second terms in the rhs of , respectively .we will apply lemma [ lemma : fano ] with and hence we choose and for some with to satisfy the power constraint .now , the first term in the rhs of is bounded as follows : where is due to lemma [ lemma : fano ] with the substitution of , and .is because for each  ] , , follows from lemma [ lemma : fano ] with , and , and is because all channel fading coefficients are assumed to be bounded away from zero and infinity .let us apply theorem [ thm : sec_rate ] with the following choice of }|v)\prod_{k\in[1:m]}p(x_k|s_k,\mathbf{h}) ] for some positive real number and positive integer specified later .we note that and \leq p ] are satisfied if where .then , the channel outputs are given as follows : }}{h_{[k-1]_n}}\right)u_k + z_2.\end{aligned}\ ] ]  because , from theorem [ thm : sec_rate ] , the following secrecy rate is achievable : to derive a lower bound on the rhs of , let us derive a lower and an upper bounds on the first and the second terms in the rhs of , respectively .we will apply lemma [ lemma : fano ] with and hence we choose and for some with to satisfy the power constraint .now , the first term in the rhs of is bounded as follows : where is due to lemma [ lemma : fano ] with the substitution of , , .next , by applying similar steps used to derive , we can show    [ [ proof - of - the - achievability - part - of - theoremthmless - for - the - case - without - the - knowledge - of - location - of - wiretapped - links ] ] proof of the achievability part of theorem [ thm : less ] for the case without the knowledge of location of wiretapped links ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~    let ) ] for some positive real number and positive integer to be specified later . to mask each element transmitted over the source - relay links in the s - bcj scheme , we generate a vector , j\in [ 1:m-1]) ] : , j\in [ 1:m-1])\\  s_k&=\begin{cases } ( v_{k , j}+l_{k , j } : j\in [ 1:m-1 ] ) , & k\in [ 1:n ] \\ ( l_{k , j } : j\in [ 1:m-1 ] ) , & k\in [ n+1:m ] \\ \end{cases}\end{aligned}\ ] ] } \mu_{k , j } ( v_{k , j}+l_{k , j})+ \sum_{j\in [ 1:n ] } \frac{\nu_j}{h_k}u_{k , j } , & k\in [ 1:n]\\ \sum_{j\in [ 1:m-1 ] } \rho_{k , j}l_{k , j } + \sum_{j\in [ 1:n ] } \frac{\nu_j}{h_k}u_{k , j } ,  & k\in [ n+1:m ] \end{cases}\end{aligned}\ ] ] where s and s are independently generated according to  ] . s are carefully chosen to cancel out the fictitious message symbols at the destination as follows .we first note that because any columns of are linearly independent , for  ] , there exists , j\in [ 1:m-1]) ] and  ] for  ] such that . to derive a lower bound on the rhs of ,let us derive a lower and an upper bounds on the first and the second terms in the rhs of , respectively .we will apply lemma [ lemma : fano ] with and hence we choose and for some with to satisfy the power constraint . then , by applying the same bounding techniques used to obtain , we can obtain next , let us derive an upper bound on the second term in the rhs of .we first fix an arbitrary  ] , , is because is independent from and , and follows from lemma [ lemma : fano ] with , and .since the above bounds do not depend on the choice of , we obtain     let ) ] for some positive real number and positive integer to be specified later .we generate a vector ) ] : )\\ s_k&=\begin{cases } ( v_{k}+u_k+l_{2k-1},u_{[k+1]_n}+l_{2k } ) , & k\in [ 1:n ] \\ ( l_{2k-1 } , l_{2k } ) , & k\in [ n+1:m ] \end{cases}\\ x_k&=\begin{cases } v_k+u_k+l_{2k-1}-\frac{h_{[k+1]_n}}{h_k}(u_{[k+1]_{n}}+l_{2k } ) , & k\in [ 1:n]\\ \rho_{2k-1}l_{2k-1}+\rho_{2k}l_{2k } , & k\in [ n+1:m ] \end{cases}\end{aligned}\ ] ] where s and s are independently generated according to  ] , there exists ) ] are chosen as }\frac{h_k}{h_j}\sigma_{2k-1|2j-1}-\sum_{k\in[1:n]}\frac{h_{[k+1]_n}}{h_j}\sigma_{2k|2j-1}\\ \rho_{2j}&=-\sum_{k\in[1:n]}\frac{h_k}{h_j}\sigma_{2k-1|2j } -\sum_{k\in[1:n]}\frac{h_{[k+1]_n}}{h_j}\sigma_{2k|2j}.\end{aligned}\ ] ] let denote the maximum of s .we note that and \leq p ] are satisfied if where   then , the channel outputs are given as follows : }}{h_{[k-1]_n}}\right)u_k  + \sum_{k=1}^{2w } \chi_k f_k+z_2,\end{aligned}\ ] ] where s are determined from s , , , and .  from theorem [ thm : sec_rate_no ], the following secrecy rate is achievable : where the maximization is over all  ] such that .note that where ) ] , relay  ] , legitimate channel output , eavesdropper channel output , and channel distribution marginalized from }p(x_k|s_k,\mathbf{h})p(y_1,y_2|x_{[1:m]},\mathbf{h},\mathbf{g})$ ] .then , theorem [ thm : sec_rate ] is immediate from . for the case without the knowledge of location of wiretapped links , theorem [ thm : sec_rate_no ] can be proved in a similar manner by assuming there are multiple eavesdroppers each of which observes every different such that .   according to the khintchine - groshev theorem of diophantine approximation , for any andalmost all except a set of lebesque measure zero , there exists a constant such that .hence , the minimum distance between the points in is bounded as follows : for almost all except a set of lebesque measure zero .now , when is known to the receiver , let denote the estimate of which is chosen as follows : then , for the choice of and , we have \\   & \overset{(a)}\leq \exp\left(-\frac{\delta^2 k_{\epsilon}^2}{8(\theta q)^{2(\tau-1 + \epsilon ) } } \right)\\  & \leq \exp\left(-\frac{\gamma^2 k_{\epsilon}^2p^{\epsilon}}{8\theta^{2(\tau-1+\epsilon)}}\right ) ,  \end{aligned}\ ] ] where is because the probability that are included in a set of lebesque measure zero is zero by assumption .let us show that the generator matrix of a reed - solomon code with block length , message length , and alphabet size satisfies the aforementioned properties .first , because is over the prime field , its element is an integer in the range , and hence the first property is satisfied . for the second property, we first note that the sum , the difference and the product over are computed by taking the modulo of the integer result . since a reed - solomon code is an mds - code , any columns of are linearly independent over , which means that the determinant of any submatrix of _ evaluated over the prime field _ is nonzero .this implies that the determinant of any submatrix of _ evaluated over real numbers _ is nonzero , and hence the second property is satisfied .s. saeedi bidokhti and g. kramer , `` capacity bounds for diamond networks with an orthogonal broadcast channel , '' _ ieee trans .inf . theory _ , submitted for publication .[ online ] .available : http://arxiv.org/abs/1510.00994 .s. e. j. richter , c. scheunert and e. a. jorswieck , `` weak secrecy in the multiway untrusted relay channel with compute - and - forward . ''_ ieee trans .inf . forens . and sec _ , vol .10 , pp . 12621273 , jun . 2015 .a. s. motahari , s. oveis - gharan , m. a. maddah - ali , and a. k. khandani , `` real interference alignment : exploiting the potential of single antenna systems , '' _ ieee trans .inf . theory _ ,60 , pp . 47994810 ,aug . 2014 .  j. xie and s. ulukus , ``secure degrees of freedom of the gaussian wiretap channel with helpers and no eavesdropper csi : blind cooperative jamming , '' in _ conference on information sciences and systems ( ciss ) _ , mar . 2013 .p. mukherjee , j. xie , and s. ulukus , `` secure degrees of freedom of one - hop wireless networks with no eavesdropper csit , '' _ ieee trans .inf . theory _ , submitted for publication .[ online ] .available : http://arxiv.org/abs/1506.06114 .s. yang , m. kobayashi , p. piantanida , and s. shamai , `` secrecy degrees of freedom of mimo broadcast channels with delayed csit , '' _ information theory , ieee transactions on _ , vol .59 , no . 9 , pp . 52445256 , 2013 .lee , w. zhao , and a. khisti , `` secure degrees of freedom of the gaussian diamond - wiretap channel , '' _ ieee trans .inf . theory _, submitted for publication .[ online ] .available : http://arxiv.org/abs/1512.06291 .a. g. davoodi and s. a. jafar , `` aligned image sets under channel uncertainty : settling a conjecture by lapidoth , shamai and wigger on the collapse of degrees of freedom under finite precision csit , '' 2014 .[ online ] .available : http://arxiv.org/abs/1403.1541 .

Abstract: in this paper , we study a diamond - relay channel where the source is connected to relays through orthogonal links and the relays transmit to the destination over a wireless multiple - access channel in the presence of an eavesdropper . the eavesdropper not only observes the relay transmissions through another multiple - access channel , but also observes a certain number of source - relay links . the legitimate terminals know neither the eavesdropper s channel state information nor the location of source - relay links revealed to the eavesdropper except the total number of such links .  for this _ wiretapped diamond - relay channel _ , we establish the optimal secure degrees of freedom . in the achievability part , our proposed scheme uses the source - relay links to transmit a judiciously constructed combination of message symbols , artificial noise symbols as well as fictitious message symbols associated with secure network coding . the relays use a combination of beamforming and interference alignment in their transmission scheme . for the converse part , we take a genie - aided approach assuming that the location of wiretapped links is known .