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Computer Science
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Construction of Z4-linear Reed-Muller codes

New quaternary Plotkin constructions are given and are used to obtain new families of quaternary codes. The parameters of the obtained codes, such as the length, the dimension and the minimum distance are studied. Using these constructions new families of quaternary Reed-Muller codes are built with the peculiarity that after using the Gray map the obtained Z4-linear codes have the same parameters and fundamental properties as the codes in the usual binary linear Reed-Muller family. To make more evident the duality relationships in the constructed families the concept of Kronecker inner product is introduced.


Introduction
In [13] Nechaev introduced the concept of Z 4 -linearity of binary codes and later, in [7], Hammons, Kumar, Calderbank, Sloane and Solé showed that several families of binary codes are Z 4 -linear. In [7] it is proved that the binary linear Reed-Muller code RM (r, m) is Z 4 -linear for r = 0, 1, 2, m − 1, m and is not Z 4 -linear for r = m − 2 (m ≥ 5). In a subsequent work, Hou, Lahtonen and Koponen [8], proved that RM (r, m) is not Z 4 -linear for 3 ≤ r ≤ m − 2.
In [7] it is introduced a construction of codes, called QRM(r, m), based on Z 4 -linear codes, such that after doing modulo two we obtain the usual binary linear Reed-Muller (RM ) codes. In [2,3] such family of codes is studied and their parameters are computed as well as the dimension of the kernel and rank. In [16] a kind of Plotkin construction was used to build a family of additive Reed-Muller codes and also in [19] it was used a Plotkin construction to obtain a sequence of quaternary linear Reed-Muller like codes. In both last quoted constructions, the images of the obtained codes under the Gray map are binary codes with the same parameters as the binary linear RM codes. Moreover, on the other hand, in [5,9,10,14] were classified all the non-equivalent Z 2 Z 4 -linear extended 1-perfect codes and their duals, the Z 2 Z 4 -linear Hadamard codes. It is a natural question to ask for the existence of families of quaternary linear codes such that, after the Gray map, the corresponding Z 4 -linear codes have the same parameters as the well known family of binary linear RM codes. In these new families, like in the usual RM (r, m) family, the code with (r, m) = (1, m) should be a Hadamard code and the code with (r, m) = (m − 2, m) should be an extended 1-perfect code.
It is well known that an easy way to built the binary RM family of codes is by using the Plotkin construction [12]. So, it seems a good matter of study to try to generalize the Plotkin construction to the quaternary linear codes and try to obtain new families of codes which contain the above mentioned Z 2 Z 4linear Hadamard codes and Z 2 Z 4 -linear extended 1-perfect codes and fulfill the same properties from a parameters point of view (length, dimension, minimum distance, inclusion and duality relationship) than the binary RM family.
In this paper we begin by studying the Z 4 -linear case and we organize it in the following way. In Section 2 we introduce the concept of quaternary code and give some constructions that could be seen as quaternary generalizations of the well known binary Plotkin construction. In Section 3, we construct several families of Z 4 -linear Reed-Muller codes and prove that they have similar parameters as the classical binary RM codes but they are not linear. In Section 4, we discuss the concept of duality for the constructed Z 4 -linear Reed-Muller codes and, finally, in Section 5 we give some conclusions and further research in the topic.

Constructions of quaternary codes 2.1 Quaternary codes
Let Z 2 and Z 4 be the ring of integers modulo two and modulo four, respectively. Let Z n 2 be the set of all binary vectors of length n and Z N 4 be the set of all quaternary vectors of length N . Any non-empty subset C of Z n 2 is a binary code and a subgroup of Z n 2 is called a binary linear code. Equivalently, any non-empty subset C of Z N 4 is a quaternary code and a subgroup of Z N 4 is called a quaternary linear code. In general, any non-empty subgroup C of Z α 2 × Z β 4 is an additive code.
The Hamming weight w H (u) of a vector in Z n 2 is the number of its nonzero coordinates. The Hamming distance d(u, v) between two vectors u, v ∈ Z n 2 is d(u, v) = w H (u − v). For quaternary codes it is more appropriate to use the Lee metric [11]. In Z 2 the Lee weight coincides with the Hamming weight, but in Z 4 the Lee weight of their elements is w L (0) = 0, w L (1) = w L (3) = 1, and w L (2) = 2. The Lee weight w L (u) of a vector in Z N 4 is the addition of the Lee weight of all the coordinates. The Lee distance d L (u, v) between two vectors is the usual Gray map, so φ(v 1 , . . . , v β ) = (ϕ(v 1 ), . . . , ϕ(v β )), and ϕ(0) = (0, 0), ϕ(1) = (0, 1), ϕ(2) = (1, 1), ϕ(3) = (1, 0). We will use the symbols 0, 1 and 2 for the all zeroes, the all ones and the all twos vectors, respectively (by the context it will be always clear we speak about the binary vectors 0, 1 or quaternary, it will also be clear the length of the vectors).
Hamming and Lee weights, as well as Hamming and Lee distances, can be generalized, in a natural way, to vectors in Z α 2 × Z β 4 by adding the corresponding weights (or distances) of the Z α 2 part and the Z β 4 part. Since C is a subgroup of Z α 2 ×Z β 4 , it is also isomorphic to an abelian structure like Z γ 2 × Z δ 4 . Therefore, we have that |C| = 2 γ 4 δ and the number of order two codewords in C is 2 γ+δ . We call such code C an additive code of type (α, β; γ, δ) and the binary image C = Φ(C) a Z 2 Z 4 -linear code of type (α, β; γ, δ). In the specific case α = 0 we see that C is a quaternary linear code and its binary image is called a Z 4 -linear code. Note that the binary length of the binary code The minimum Hamming distance d of a Z 2 Z 4 -linear code C is the minimum value of d(u, v), where u, v ∈ C and u = v. Notice that the Hamming distance of a Z 2 Z 4 -linear code C coincides with the Lee distance defined in the additive code C = Φ −1 (C). From now on, when we work with distances it must be understood that we deal with Hamming distances in the case of binary codes or Lee distances in the case of additive codes.
Although C could not have a basis, it is appropriate to define a generator matrix for C as where B 2 is a γ × α matrix; Q 2 is a γ × β matrix; B 1 is a δ × α matrix and Q 1 is a δ × β matrix. Matrices B 1 , B 2 are binary and Q 1 , Q 2 are quaternary, but the entries in Q 2 are only zeroes or twos. Two additive codes C 1 and C 2 both of the same length are said to be monomially equivalent, if one can be obtained from the other by permuting the coordinates and multiplying by −1 of certain coordinates. Additive codes which differ only by a permutation of coordinates are said to be permutation equivalent.
For Z 2 Z 4 -linear codes is usual to use the following definition of inner product in Z α 2 × Z β 4 that we will call the standard inner product [18,4]: where u, v ∈ Z α 2 × Z β 4 . We can also write the standard inner product as where J N = 2I α 0 0 I β , N = α + β, is a diagonal matrix over Z 4 . Note that when α = 0 the inner product is the usual one for vectors over Z 4 and when β = 0 it is twice the usual one for vectors over Z 2 .
For α = 0 and N = β = 2 i , i = 1, 2, 3, . . ., we can define the inner product in an alternative way. Let K 2 = 1 0 0 3 be a matrix over Z 4 and define where denotes the Kronecker product of matrices. We call the Kronecker inner product the following: The additive dual code of C, denoted by C ⊥ , is defined in the standard way as for all v ∈ C} or, using the Kronecker inner product The definition and notations will be the same for the Z 4 -duality obtained by using the standard inner product or the Kronecker inner product and the difference will be clear from the context.
Note that u, v ⊗N = u·K N ·v t = u, v·K N . Hence, both additive dual codes by using the standard inner product or the Kronecker inner product, respectively, are monomially equivalent and so they have the same weight distribution. For both inner products, the additive dual code C ⊥ is also an additive code, that is a subgroup of Z α 2 ×Z β 4 . Its weight enumerator polynomial is related to the weight enumerator polynomial of C by the MacWilliams identity [6]. The corresponding binary code Φ(C ⊥ ) is denoted by C ⊥ and called the Z 2 Z 4 -dual code of C. In the case α = 0, the code C ⊥ is also called the quaternary dual code of C and C ⊥ the Z 4 -dual code of C. Notice that C and C ⊥ are not dual in the binary linear sense but the weight enumerator polynomial of C ⊥ is the McWilliams transform of the weight enumerator polynomial of C. Given an additive code C of type (α, β, γ, δ) it is known the type of the additive dual code ( [4] for additive codes with α = 0 and [7] for additive codes with α = 0).
In the present paper, as we will see later, the duality concept using the Kronecker inner product will make more visible the property that if a code C belongs to a family of Reed-Muller codes then its dual code belongs to the same family.
From now on, we focus our attention specifically to additive codes with α = 0, so quaternary linear codes such that after the Gray map they give rise to Z 4 -linear codes. Given a quaternary linear code of type (0, β; γ, δ), we will write (N ; γ, δ) to say that α = 0 and β = N .

The Plotkin construction
In this section, we show that the well-known binary Plotkin construction can be generalized to quaternary linear codes.
Let A and B be two quaternary linear codes of types (N ; γ A , δ A ) and (N ; γ B , δ B ) and minimum distances d A , d B , respectively. Given u ∈ Z N 4 define supp(u) ⊂ {1, . . . , N } as the set of nonzero coordinates of vector u.
Definition 1 (Plotkin Construction) Given two quaternary linear codes A and B, we define a quaternary linear code as It is easy to see that if G A and G B are generator matrices of A and B, respectively, then the matrix is a generator matrix of the code PC(A, B).

Proposition 2
The quaternary linear code PC(A, B) defined using the Plotkin construction is of type (2N ; γ, δ), where γ = γ A + γ B and δ = δ A + δ B ; the binary length is n = 4N ; the size is 2 γ+2δ and the minimum distance is d = min{2d A , d B }.
Proof: The type, the binary length and the size of PC(A, B) can be easily computed from the definition of the code. The minimum distance can be established as in the binary case [12] but, by completeness, we include the proof. Let us consider any vector u ∈ PC(A, B) such that u = (u 1 |u 1 + u 2 ), where u 1 ∈ A and u 2 ∈ B. Since PC(A, B) is a quaternary linear code, it is enough to prove that the weight w L (u) is not less than d. If If u 2 = 0, by using the triangle inequality we immediately obtain Hence d ≥ min{2d A , d B }. The equality holds because taking the specific vectors u 1 ∈ A with minimum weight d A and u 2 ∈ B with minimum weight d B we obtain w L (u 1 |u 1 ) = 2d A and w L (0|v 2 ) = d B .

The quaternary Plotkin construction
A useful generalization of the above construction to obtain quaternary linear codes is the following construction, called the quaternary Plotkin construction. Such construction was used, for example, in [10] for the classification of all Z 4 -linear Hadamard codes.

Definition 3 (Quaternary Plotkin Construction) Given two quaternary linear codes A and B, we define the quaternary linear code
It is easy to see that if G A and G B are generator matrices of A and B, then the matrix is a generator matrix of the code QP(A, B).

Proposition 4 The quaternary linear code QP(A, B) given in Definition 3 is of type
Proof: The type, the binary length and the size of QP(A, B) can be easily computed from the definition of the code. To check the minimum distance of QP(A, B) let us consider any vector u ∈ QP(A, B). Vector u can be represented by u = (u 1 |u 1 |u 1 |u 1 ) + (0|u 2 |2u 2 |3u 2 ), where u 1 ∈ A and u 2 ∈ B. Since QP(A, B) is a quaternary linear code it is enough to show that the weight of u is at least d. If The equality holds taking a vector u 1 ∈ A of minimum weight. For It is easy to see that if G A , G B , G C and G D are generator matrices of A, B, C and D, then the matrix is a generator matrix of the code DP(A, B, C, D).

Proposition 6
The quaternary linear code DP(A, B, C, D) given in Defini- the binary length is n = 8N ; the size is 2 γ+2δ and the minimum distance is Proof: The type, the binary length and the size of the code DP(A, B, C, D) can be easily computed from the definition.
To check the minimum distance of the code DP(A, B, C, D) let us consider any vector u from this code. It can be represented as u = (u 1 |u 1 |u 1 |u 1 ) + (0|u 2 |2u 2 |3u 2 ) + (0|0|u 3 |u 3 ) + (0|0|0|u 4 ), where u 1 ∈ A, u 2 ∈ B, u 3 ∈ C and u 4 ∈ D. Since DP(A, B, C, D) is a quaternary linear code it is enough to show that the weight of u is, at least, d.
If u 2 = 0 then we can write u = (u 1 |u 1 |u 1 |u 1 ) + (0|0|u 3 |u 3 + u 4 ) so that u ∈ PC((A|A), PC(C, D)), where (A|A) is the code generated by (G A |G A ). Using Proposition 2 we obtain Note that in case B = C the bound is tight because d B = d C and the minimum distance d = min{4d A , 2d C , d D } can be obtained taking specific vectors from A, C or D.

The BQ-Plotkin construction
We slightly change the construction given in Definition 5 in order to obtain a tight bound for the minimum distance. We call this new construction the BQ-Plotkin construction.
Let A, B and C be three quaternary linear codes of types (N ; γ A , δ A ), (N ; γ B , δ B ), (N ; γ C , δ C ), with minimum distances d A , d B and d C , respectively.
Definition 7 (BQ-Plotkin Construction) Let G A , G B and G C be generator matrices of the quaternary linear codes A, B and C, respectively. We define a new code BQ(A, B, C) as the quaternary linear code generated by where G ′ B is the matrix obtained from G B after switching twos by ones in their γ B rows of order two andĜ B is the matrix obtained from G B after removing their γ B rows of order two.
Proof: The type, the length and the size of BQ(A, B, C) can be easily computed from the definition of the code.
If u 2 = 0 then by using the same arguments as in Proposition 6 we have If u 2 = 0 then we distinguish two cases. If u 4 = 0 then Using twice the triangle inequality, the case u 4 = 0 easily gives But the equality holds after the following considerations.
Taking the specific vector u 1 ∈ A with minimum weight d A we obtain w L (u 1 |u 1 |u 1 |u 1 ) = 4d A .
Taking the specific vector u 4 ∈ C with minimum weight d C we obtain w L (0|0|0|u 4 ) = d C .

Quaternary Reed-Muller codes
The usual binary linear RM family of codes is one of the oldest and interesting family of codes. The codes in this family are easy to decode and their combinatorial properties are of great interest to produce new optimal codes. For any integer m ≥ 1 the family of binary linear RM codes is given by the sequence RM (r, m), where 0 ≤ r ≤ m. The code RM (r, m) is called the r-th order binary linear Reed-Muller code of length n = 2 m and it is true that Let 0 ≤ r ≤ m, m ≥ 1. Following [12] the RM (r, m) code of order r can be constructed by using the Plotkin construction in the following way: It is important to note that if we fix m, once we know the sequence RM (r, m) for all 0 ≤ r ≤ m, then it is easy to obtain the new sequence RM (r, m + 1) by using the Plotkin construction (3).
Moreover, the codes in the RM family fulfill the basic properties summarized in the following theorem: In the recent literature [7,20,2,3] several families of quaternary linear codes have been proposed and studied trying to generalize the RM codes. However, when we take the corresponding Z 4 -linear codes, they do not satisfy all the properties in Theorem 9. This last requirement is the main goal of the present work, to construct new families of quaternary linear codes such that, after the Gray map, we obtain Z 4 -linear codes with the parameters and properties quoted in Theorem 9. The result of the present paper generalizes the results in [19].

The family of RM(r, 1) codes
We begin by considering the trivial case of m = 1, that is, the case of codes of binary length n = 2 1 . The quaternary linear Reed-Muller code RM(0, 1) is the repetition code with only one nonzero codeword (the vector with only one quaternary coordinate of value 2). This quaternary linear code is of type (1; 1, 0). The code RM(1, 1) is the whole space Z 1 4 , so a quaternary linear code of type (1; 0, 1).
These two codes, RM(0, 1) and RM(1, 1), after the Gray map, give binary codes with the same parameters of the corresponding binary codes RM (r, 1) and with the same properties described in Theorem 9. In this case, when m = 1, not only these codes have the same parameters, but they have the same codewords.
We will refer to these codes as RM 0 (0, 1) and RM 0 (1, 1), respectively, as it is shown in Table 1. In each entry of this table there are the parameters (γ, δ) of the corresponding code of type (N ; γ, δ).
Since we will need an specific representation for these codes in Table 1, we will agree in using further the following matrices as the generator matrices for each one of them. The generator matrix of RM 0 (0, 1) is G 0 (0, 1) = 2 and the generator matrix of RM 0 (1, 1) is G 0 (1, 1) = 1 .

Plotkin and BQ-Plotkin constructions
The first important point is to apply the Plotkin construction to quaternary linear Reed-Muller codes.
For m = 3, it is well known that there exist two Z 4 -linear Hadamard codes [10]. So, our goal is to construct two families of quaternary linear Reed-Muller codes as it is shown in Table 3. The codes in the first row of Table 3 can be obtained due to the Plotkin construction from the codes of Table 2. But, the codes in the second row can not be obtained by using only the Plotkin construction. It is in this case that we need to exploit the new BQ-Plotkin construction as we will see later in this section.
Constructions of additive codes with the parameters of the binary linear Reed-Muller codes by using only the Plotkin construction were initiated in [16,19].   the codes generated by G ′ i andĜ i , respectively, such that for i = 1, 2, 3 it is true that
Proof: It is straightforward to see that the Plotkin construction fulfills the properties.
For the BQ-Plotkin construction the property (i) is clear from Definition 7. Now, the generator matrix ofBQ(A i+2 , A i+1 , A i ) has the following form: Using the properties (ii) and (iii) for the matrices using the properties (ii) and (iii) for the matrices G i , G i+1 , G i+2 we obtain the property (iii) for BQ(A i+2 , A i+1 , A i ) and i = 1.
Finally, the generator matrix of BQ(A i+2 , A i+1 , A i ) has the form Using the properties (ii), (iii) and (iv) for the matrices G i , G i+1 , G i+2 we obtain the property (iv) for BQ(A i+2 , A i+1 , A i ) and i = 1.  Proof: The type (4N ; γ s r,m , δ s r,m ) of the code RM s (r, m) and the minimum distance can be computed from Proposition 8.
Proof: All the rows in matrix G s (m − 1, m) are vectors of even weight. So, to prove that this matrix generates RM s (m − 1, m) we only need to check if the dimension is the adequate.
We will prove, by induction on m ≥ 1, that γ s m−1,m = 1 and δ s m−1,m = 2 m−1 − 1. The claim is trivially true for m = 1 and m = 2 using the matrices defined in section 3.1 and in (4). Suppose the claim is true for m ≥ 2.  Using Theorems 10 and 12 we can construct the RM codes in the two rows of Table 3. We do not write the generator matrices for codes RM 0 (r, 3) because they can be directly obtained from the respective codes for m = 2 by using the Plotkin construction. For the codes in the family RM 1 (r, 3) we present the generator matrices as a direct application of Theorem 12: and the remaining code RM 1 (3, 3) in the family is the whole space Z 2 2 4 . All these codes, after the Gray map, give binary codes with the same parameters as the codes RM (r, 3) and with the same properties described in Theorem 9. In the case m = 3, like in the case m = 2 not only these codes have the same parameters, but they have the same codewords. This is not in this way for all the other values m > 3. Now, from Table 3 and by using the Plotkin construction we can construct the two families of the codes RM s (r, 4) for s = 0, 1, as it is shown in Table 4. Note that the family of codes RM 1 (r, 4) also can be obtained using the BQ-Plotkin construction from the family of codes RM 0 (r, 2) in Table 2.
Note that RM 0 (r, 5) only can be obtained applying the Plotkin construction, RM 2 (r, 5) only can be obtained applying the BQ-Plotkin construction, but RM 1 (r, 5) can be obtained by using the Plotkin or the BQ-Plotkin construction.
In general, for m > 1, the code RM 0 (r, m) can be only obtained applying the Plotkin construction. As it is proved in Theorems 10 and 12 the constructed families of RM codes satisfy the same properties we stated for binary linear Reed-Muller codes in Theorem 9 except for the duality. In the following Section we will discuss this topic.
Notice that, after the Gray map, the constructed RM families of quaternary linear Reed-Muller codes have not only the same parameters as the usual binary linear family of RM codes, but also the characteristic properties of codes RM s (1, m) and RM s (m − 2, m) as it is stated in the following proposition.

Duality
For the usual binary linear RM codes we know that the codes RM (r, m) and RM (m − r − 1, m) are dual to each other. The families of RM codes have the same property if we use the Kronecker inner product to define the Z 4 -duality.
Throughout this section the notion of duality will be related to the Kronecker inner product defined in (2).
We begin by studying the duality properties for the family of RM codes obtained by using the BQ-Plotkin construction, that is, we are going to prove the duality relationships for the family of codes RM s (r, m), m odd, s = m−1 2 , constructed from the family RM s−1 (r, m − 2).
Basically, we will prove this fact by induction but, previously, we need two technical lemmas. We will use G s (r, m) to refer to the generator matrix of code RM s (r, m); the matrices G ′ s (r, m),Ĝ s (m − r − 1, m) will have the meaning that we introduced in Definition 7 and RM ′ s (r, m),R M s (m − r − 1, m) will be the codes generated by G ′ s (r, m) andĜ s (m − r − 1, m), respectively.
Lemma 17 Let u, v ∈ Z N 4 be any two vectors such that u = (u 1 |u 2 ) and Proof: Straightforward from the Kronecker inner product definition. Proof: We proceed by induction on m beginning with m = 3. Using (9) it is easy to see that the assertion is true for m = 3.
For the case when r is odd the statement is trivially true, since there is nothing to proof. Indeed, from Lemma 15, γ s r,m = 0 and so G ′ s (r, m)\Ĝ s (r, m) = ∅. Hence, along this proof we can take r as an even integer. Now, for m > 3 and 0 < r ≤ m − 2, assume by induction hypothesis that for all x ∈ G ′ s−1 (r, m − 2)\Ĝ s−1 (r, m − 2) and y ∈R M s (m − r − 3, m − 2) is x, y ⊗N/4 = 0.
Finally, we prove the statement for r = 0; r = m − 1 and r = m. For r = 0, we proceed by induction. Case m = 3 is trivially true taking into account (9). We have u ∈ G ′ s (0, m)\Ĝ s (0, m) and so u is the all ones vector u = (1, 1, . . . Proof: Since |RM s (r, m)| · |RM s (m − r − 1, m)| = 2 m , it suffices to prove that for every u ∈ G s (r, m) and v ∈ G s (m − r − 1, m) we have u, v ⊗N = 0. We proceed by induction on m. The claim is trivially true for m = 1. Now, for m > 1 and 0 ≤ r ≤ m, assume by induction hypothesis that for all x ∈ G s−1 (r, m − 2) and y ∈ G s−1 (m − r − 3, m − 2) it is true that x, y ⊗N/4 = 0.

Conclusion
New constructions based on quaternary linear codes has been proposed such that, after doing a Gray map, the obtained Z 4 -linear codes fulfill the same properties and fundamental characteristics as the binary linear RM codes. Apart from the parameters characterizing each code an important property which remains in these new presented families is that the first order RM code is, under the Gray map, a Z 4 -linear Hadamard code and the (m − 2)-th order RM code, after the Gray map, is a Z 4 -linear extended perfect code, like in the usual binary case. So the families of codes obtained in the paper, after the Gray map, contain the families of Z 4 -linear extended perfect and Z 4 -linear Hadamard codes introduced in [9,10]. Moreover, it is important to note that, after defining the Kronecker inner product, the codes RM(r, m) and RM(m − r − 1, m) are dual each other like in the binary linear case.
There are several questions and subjects related to this work where would be of great interest to go deeply. The first one is about the generalization of the constructions of RM codes to the case of general additive codes, so the case of additive codes with α = 0. It is known from [5] that there exist Z 2 Z 4 -linear perfect and Hadamard codes (which are not Z 4 -linear) and these could be the starting point of the new families. Other questions of interest are related to uniqueness (up to equivalence) of the codes in a given family of RM codes, weight distribution, etc.