$${\mathbb{Z}_2\mathbb{Z}_4}$$ -linear codes: rank and kernel

Des. Codes Cryptogr. (2010) 56:43–59 DOI 10.1007/s10623-009-9340-9 Z2Z4 -linear codes: rank and kernel Cristina Fernández-Córdoba · Jaume Pujol · Mercè Villanueva Received: 25 February 2009 / Revised: 2 September 2009 / Accepted: 19 October 2009 / Published online: 6 November 2009 © Springer Science+Business Media, LLC 2009 Abstract A code C is Z2 Z4 -additive if the set of coordinates can be partitioned into two subsets X and Y such that the punctured code of C by deleting the coordinates outside X (respectively, Y ) is a binary linear code (respectively, a quaternary linear code). The corresponding binary codes of Z2 Z4 -additive codes under an extended Gray map are called Z2 Z4 -linear codes. In this paper, the invariants for Z2 Z4 -linear codes, the rank and dimension of the kernel, are studied. Specifically, given the algebraic parameters of Z2 Z4 -linear codes, the possible values of these two invariants, giving lower and upper bounds, are established. For each possible rank r between these bounds, the construction of a Z2 Z4 -linear code with rank r is given. Equivalently, for each possible dimension of the kernel k, the construction of a Z2 Z4 -linear code with dimension of the kernel k is given. Finally, the bounds on the rank, once the kernel dimension is fixed, are established and the construction of a Z2 Z4 -linear code for each possible pair (r, k) is given. Keywords Quaternary linear codes · Z4 -linear codes · Z2 Z4 -additive codes · Z2 Z4 -linear codes · Kernel · Rank Communicated by Victor A. Zinoviev. The material in this paper was presented in part at the XI International Symposium on Problems of Redundancy in Information and Control Systems, Saint Petersburg, Russia, July 2007 [13]; and at the 2nd International Castle Meeting on Coding Theory and Applications, Medina del Campo, Spain, September 2008 [14]. C. Fernández-Córdoba (B) · J. Pujol · M. Villanueva Department of Information and Communications Engineering, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain e-mail: cristina.fernandez@autonoma.edu J. Pujol e-mail: jaume.pujol@autonoma.edu M. Villanueva e-mail: merce.villanueva@autonoma.edu 123 44 C. Fernández-Córdoba et al. Mathematics Subject Classification (2000): 94B60 · 94B25 1 Introduction Let Z2 and Z4 be the ring of integers modulo 2 and modulo 4, respectively. Let Zn2 be the set of all binary vectors of length n and let Zn4 be the set of all n-tuples over the ring Z4 . In this paper, the elements of Zn4 will also be called quaternary vectors of length n. Any nonempty subset C of Zn2 is a binary code and a subgroup of Zn2 is called a binary linear code or a Z2 -linear code. Equivalently, any nonempty subset C of Zn4 is a quaternary code and a subgroup of Zn4 is called a quaternary linear code. Quaternary codes can be viewed as binary codes under the usual Gray map defined as φ(0) = (0, 0), φ(1) = (0, 1), φ(2) = (1, 1), φ(3) = (1, 0) in each coordinate. If C is a quaternary linear code, then the binary code C = φ(C ) is called a Z4 -linear code. The dual of a quaternary linear code C , denoted by C ⊥ , is called the quaternary dual code and is defined in the standard way [19] in terms of the usual inner product for quaternary vectors [15]. The binary code C⊥ = φ(C ⊥ ) is called the Z4 -dual code of C = φ(C ). Since 1994, quaternary linear codes have became significant due to its relationship to some classical well-known binary codes as the Nordstrom-Robinson, Kerdock, Preparata, Goethals or Reed-Muller codes [15]. It was proved that the Kerdock code and some Preparata-like codes are Z4 -linear codes and, moreover, the Z4 -dual code of the Kerdock code is a Preparata-like code. Lately, more families of quaternary linear codes, called Q R M, Z R M and RM, related to the Reed-Muller codes have been studied in [3,4,25], respectively. Additive codes were first defined by Delsarte in 1973 in terms of association schemes [11,12]. In general, an additive code, in a translation association scheme, is defined as a subgroup of the underlying Abelian group. In the special case of a binary Hamming scheme, that is, when the underlying Abelian group is of order 2n , the only structures for the Abelian β group are those of the form Zα2 × Z4 , with α + 2β = n. Therefore, the subgroups C of β Zα2 × Z4 are the only additive codes in a binary Hamming scheme. In order to distinguish them from additive codes over finite fields [2], we will hereafter call them Z2 Z4 -additive codes [5,9,22]. The Z2 Z4 -additive codes are also included in other families of codes with an algebraic structure, such as mixed group codes [18] and translation invariant propelinear codes [24]. β β Let C be a Z2 Z4 -additive code, which is a subgroup of Zα2 × Z4 . Let  : Zα2 × Z4 −→ Zn2 , where n = α + 2β, be an extension of the usual Gray map given by (x, y) = (x, φ(y1 ), . . . , φ(yβ )) β for any x ∈ Zα2 , and any y = (y1 , . . . , yβ ) ∈ Z4 . This Gray map is an isometry which transforms Lee distances defined in a Z2 Z4 -additive code β C over Zα2 × Z4 to Hamming distances defined in the corresponding binary code C = (C ). The binary code C = (C ) is called Z2 Z4 -linear. Note that Z2 Z4 -linear codes include binary linear codes and Z4 -linear codes. Two binary codes C1 and C2 of length n are said to be isomorphic if there exists a coordinate permutation π such that C2 = {π(c) | c ∈ C1 }. Two Z2 Z4 -additive codes C1 and C2 are said to be monomially equivalent, if one can be obtained from the other by permuting the coordinates and (if necessary) changing the signs of certain Z4 coordinates. They are said to be permutation equivalent if they differ only by a permutation of coordinates [16]. Note that if two Z2 Z4 -additive codes C1 and C2 are monomially equivalent, then, after the Gray map, 123 Z2 Z4 -linear codes: rank and kernel 45 the corresponding Z2 Z4 -linear codes C1 = (C1 ) and C2 = (C2 ) are isomorphic as binary codes. Two structural properties of nonlinear binary codes are the rank and dimension of the kernel. The rank of a binary code C, rank(C), is simply the dimension of C, which is the linear span of the codewords of C. The kernel of a binary code C, K (C), is the set of vectors that leave C invariant under translation, i.e. K (C) = {x ∈ Zn2 | C + x = C}. If C contains the all-zero vector, then K (C) is a binary linear subcode of C. In general, C can be written as the union of cosets of K (C), and K (C) is the largest such linear code for which this is true [1]. We will denote the dimension of the kernel of C by ker (C). The rank and dimension of the kernel have been studied for some families of Z2 Z4 -linear codes [3,8,7,17,20,21,23]. These two parameters do not always give a full classification of Z2 Z4 -linear codes, since two nonisomorphic Z2 Z4 -linear codes could have the same rank and dimension of the kernel. In spite of that, they can help in classification, since if two Z2 Z4 -linear codes have different ranks or dimensions of the kernel, they are nonisomorphic. Moreover, in this case the corresponding Z2 Z4 -additive codes are not monomially equivalent, so these two parameters can also help to distinguish between Z2 Z4 -additive codes that are not monomially equivalent. Most of the concepts on Z2 Z4 -additive codes have been implemented recently as a new package in Magma, including the computation of the rank and kernel, and the construction of some families of Z2 Z4 -additive codes [6,10]. The aim of this paper is the study of the rank and dimension of the kernel of Z2 Z4 -linear codes, given their algebraic parameters. The paper is organized as follows. In Sect. 2, we give some properties related to both Z2 Z4 -additive and Z2 Z4 -linear codes, including the linearity of Z2 Z4 -linear codes. In Sect. 3, we determine all possible values of the rank for Z2 Z4 -linear codes and we prove the existence of a Z2 Z4 -linear code with rank r for all possible values of r . Equivalently, in Sect. 4, we establish all possible values of the dimension of the kernel for Z2 Z4 -linear codes and we prove the existence of a Z2 Z4 -linear code with dimension of the kernel k for all possible values of k. In Sect. 5, we determine all possible pairs of values (r, k) for which there exist a Z2 Z4 -linear code with rank r and dimension of the kernel k and we construct a Z2 Z4 -linear code for any of these possible pairs. Finally, the conclusions are given in Sect. 6. 2 Preliminaries β Let C be a Z2 Z4 -additive code. Since C is a subgroup of Zα2 × Z4 , it is also isomorphic to γ an Abelian structure Z2 × Zδ4 . Therefore, C is of type 2γ 4δ as a group, it has |C | = 2γ +2δ codewords and the number of order two codewords in C is 2γ +δ . Let X (respectively Y ) be the set of Z2 (respectively Z4 ) coordinate positions, so |X | = α and |Y | = β. Unless otherwise stated, the set X corresponds to the first α coordinates and Y corresponds to the last β coordinates. Call C X (respectively CY ) the punctured code of C by deleting the coordinates outside X (respectively Y ). Let Cb be the subcode of C which contains all order two codewords and let κ be the dimension of (Cb ) X , which is a binary linear code. For the case α = 0, we will write κ = 0. Considering all these parameters, we will say that C (or equivalently C = (C )) is of type (α, β; γ , δ; κ). Although a Z2 Z4 -additive code C is not a free module, every codeword is uniquely expressible in the form 123 46 C. Fernández-Córdoba et al. γ  λi u i + δ  µjvj, j=1 i=1 β where λi ∈ Z2 for 1 ≤ i ≤ γ , µ j ∈ Z4 for 1 ≤ j ≤ δ and u i , v j are vectors in Zα2 × Z4 of order two and four, respectively. The vectors u i , v j give us a generator matrix G of size (γ + δ) × (α + β) for the code C . Moreover, we can write G as   B1 2B3 G= , (1) B2 Q where B1 , B2 are matrices over Z2 of size γ × α and δ × α, respectively; B3 is a matrix over Z4 of size γ × β with all entries in {0, 1} ⊂ Z4 ; and Q is a matrix over Z4 of size δ × β with quaternary row vectors of order four. Let In be the identity matrix of size n × n. In [15], it was shown that any quaternary linear code of type 2γ 4δ is permutation equivalent to a quaternary linear code with a generator matrix of the form   2T 2Iγ 0 GS = , (2) S R Iδ where R, T are matrices over Z4 with all entries in {0, 1} ⊂ Z4 , and of size δ × γ and γ × (β − γ − δ), respectively; and S is a matrix over Z4 of size δ × (β − γ − δ). The following theorem is a generalization of this result for Z2 Z4 -additive codes, so it gives a canonical generator matrix for these codes. Theorem 1 [5] Let C be a Z2 Z4 -additive code of type (α, β; γ , δ; κ). Then, C is permutation equivalent to a Z2 Z4 -additive code with canonical generator matrix of the form ⎞ ⎛ Iκ T ′ 2T2 0 0 (3) G S = ⎝ 0 0 2T1 2Iγ −κ 0 ⎠ , 0 S′ S R Iδ where T ′ , S ′ are matrices over Z2 ; T1 , T2 , R are matrices over Z4 with all entries in {0, 1} ⊂ Z4 ; and S is a matrix over Z4 . The concept of duality for Z2 Z4 -additive codes was also studied in [5], where the approβ priate inner product for any two vectors u, v ∈ Zα2 × Z4 was defined. Actually, in [5] it was shown that, given a finite Abelian group, the inner product is uniquely defined after fixing the generators in each one of the Abelian elementary groups in its decomposition. In our case, β the inner product in Zα2 × Z4 is defined over Z4 as u·v =2 α  u i vi + α+β  u j v j ∈ Z4 , j=α+1 i=1 β where u, v ∈ Zα2 × Z4 and the computations are made taking the zeros and ones in the first α coordinates as quaternary zeros and ones, respectively. If α = 0, the inner product is the usual one for quaternary vectors, and if β = 0, it is twice the usual one for binary vectors. Then, the additive dual code of C , denoted by C ⊥ , is defined in the standard way β C ⊥ = {v ∈ Zα2 × Z4 | u · v = 0 for all u ∈ C }. 123 Z2 Z4 -linear codes: rank and kernel 47 The corresponding binary code (C ⊥ ) is denoted by C⊥ and called the Z2 Z4 -dual code of C. Moreover, in [5] it was proved that the additive dual code C ⊥ , which is also a Z2 Z4 -additive code, is of type (α, β; γ̄ , δ̄; κ̄), where γ̄ = α + γ − 2κ, δ̄ = β − γ − δ + κ, κ̄ = α − κ. (4) The following two lemmas are a generalization of the same results proved for quaternary vectors and quaternary linear codes, respectively, in [15]. Let u∗v denote the component-wise β product for any u, v ∈ Zα2 × Z4 . β Lemma 1 For all u, v ∈ Zα2 × Z4 , we have (u + v) = (u) + (v) + (2u ∗ v). Proof Straightforward using the same arguments as for quaternary vectors to prove that for β ⊔ ⊓ all u, v ∈ Z4 , (u + v) = (u) + (v) + (2u ∗ v), [15], [26, p. 45]. β Note that if u or v are vectors in Zα2 × Z4 of order two, then (u + v) = (u) + (v). Lemma 2 Let C be a Z2 Z4 -additive code. The Z2 Z4 -linear code C = (C ) is a binary linear code if and only if 2u ∗ v ∈ C for all u, v ∈ C . Proof Straightforward by Lemma 1 and using the same arguments as for quaternary linear codes [15,26]. ⊔ ⊓ γ Note that if G is a generator matrix of a Z2 Z4 -additive code C as in (1) and {u i }i=1 and {v j }δj=1 are the row vectors of order two and four in G , respectively, then the Z2 Z4 -linear code C = (C ) is a binary linear code if and only if 2v j ∗ vk ∈ C , for all j, k satisfying 1 ≤ j < k ≤ δ, since the component-wise product is bilinear. 3 Rank of Z2 Z4 -additive codes Let C be a Z2 Z4 -additive code of type (α, β; γ , δ; κ) and let C = (C ) be the corresponding Z2 Z4 -linear code of binary length n = α + 2β. In this section, we will study the rank of these Z2 Z4 -linear codes C. We will show that there exists a Z2 Z4 -linear code C of type (α, β; γ , δ; κ) with r = rank(C) for any possible value of r . Lemma 3 Let C be a Z2 Z4 -additive code of type (α, β; γ , δ; κ) and let C = (C ) be the γ corresponding Z2 Z4 -linear code. Let G be a generator matrix of C as in (1) and let {u i }i=1 δ be the rows of order two and {v j } j=1 the rows of order four in G . Then, C is generated by γ {(u i )}i=1 , {(v j ), (2v j )}δj=1 and {(2v j ∗ vk )}1≤ j<k≤δ . Proof If x ∈ C , then x can be expressed as x = v j1 + · · · + v jm + w, where { j1 , . . . , jm } ⊆ {1, . . . , δ} and w is a codeword of order two. By Lemma 1, (x) = (v j1 +· · ·+v jm )+(w), γ where (w) is a linear combination of {(u i )}i=1 and {(2v j )}δj=1 , and (v j1 + · · · + v jm ) = (v j1 ) + · · · + (v jm ) + 1≤k<l≤m (2v jk ∗ v jl ). Therefore, (x) is generated by γ ⊔ ⊓ {(u i )}i=1 , {(v j ), (2v j )}δj=1 and {(2v j ∗ vk )}1≤ j<k≤δ . 123 48 C. Fernández-Córdoba et al. Proposition 1 Let C be a Z2 Z4 -linear code of binary length n = α + 2β and type (α, β; γ , δ; κ). Then,    δ . rank(C) ∈ γ + 2δ, . . . , min β + δ + κ, γ + 2δ + 2 Proof Let G S be a canonical generator matrix of C = −1 (C) as in (3). In the generator γ matrix G S there are γ rows of order two, {u i }i=1 , and δ rows of order four, {v j }δj=1 . Then, by γ Lemma 3, we can take the matrix G whose row vectors are {(u i )}i=1 , {(v j ), (2v j )}δj=1 and {(2v j ∗ vk )}1≤ j<k≤δ , as a generator matrix of C. γ The binary vectors {(u i )}i=1 and {(v j ), (2v j )}δj=1 are linearly independent over Z2 . Thus, rank(C) = γ + 2δ + r̄ , where r̄ is the number of additional   independent vectors taken from {(2v j ∗ vk )}1≤ j<k≤δ . Note that there are at most 2δ of such vectors.  Using row reduction in −1 (G), the 2δ vectors {2v j ∗ vk }1≤ j<k≤δ can be transformed into vectors with zeroes in the last γ − κ + δ coordinates. Therefore, there are at most  min β − (γ − κ) − δ, 2δ of such additional independent vectors, so the upper bound of    the rank is min β + δ + κ, γ + 2δ + 2δ . The lower bound follows from the case where the code C is both binary linear and Z2 Z4 linear. ⊔ ⊓ Let C be a Z2 Z4 -additive code oftype (α, β; γ , δ; κ) and C = (C ) with rank(C) =    . Let G be a generator matrix γ + 2δ + r̄ , where r̄ ∈ 0, . . . , min β − (γ − κ) − δ, 2δ γ of C as in (1), where {u i }i=1 are the rows of order two and {v j }δj=1 the rows of order four. γ By the proof of Proposition 1, the Z2 Z4 -additive code SC generated by {u i }i=1 , {v j }δj=1 and {2v j ∗ vk }1≤ j<k≤δ is of type (α, β; γ + r̄ , δ; κ) and it is easy to check that (SC ) = C, by Lemma 3. The following theorem follows. Theorem 2 Let C be a Z2 Z4 -linear code. Then, C is both binary linear and Z2 Z4 -linear. For the parameters α, β, γ , δ, κ given by some families of Z2 Z4 -linear codes such as, for example, extended 1-perfect Z2 Z4 -linear codes ([7,22] or Example 2), the upper bound above is tight. We also know Z2 Z4 -linear codes such that the rank is in between these two bounds such as, for example, the Hadamard Z4 -linear codes ([23] or Example 1). Example 1 For any integer t ≥ 3 and each δ ∈ {1, . . . , ⌊(t + 1)/2⌋} there exists a unique (up to isomorphism) extended 1-perfect Z4 -linear code C of binary length n = 2t , such that the Z4 -dual code of C is of type (0, β; γ , δ), where β = 2t−1 and γ = t + 1 − 2δ [17]. The Hadamard Z4 -linear codes H are the Z4 -dual of the extended 1-perfect Z4 -linear codes. The rank of Hadamard Z4 -linear codes was computed in [23] and the rank of extended 1-perfect Z4 -linear codes in [7,17]. Specifically,   if δ ≥ 3 γ + 2δ + δ−1 2 rank(H ) = γ + 2δ if δ = 1, 2 and rank(C) = γ̄ + 2δ̄ + δ = β + δ̄ (except when t = 4 and δ = 1), where γ̄ = γ and δ̄ = β −γ −δ by (4) taking α = 0 = κ. Note that the rank of the extended 1-perfect Z4 -linear codes satisfies the upper bound. Example 2 For any integer t ≥ 3 and each δ ∈ {0, . . . , ⌊t/2⌋} there exists a unique (up to isomorphism) extended 1-perfect Z2 Z4 -linear code C of binary length n = 2t , such that the 123 Z2 Z4 -linear codes: rank and kernel 49 Z2 Z4 -dual code of C is of type (α, β; γ , δ) with α  = 0, where α = 2t−δ , β = 2t−1 − 2t−δ−1 and γ = t + 1 − 2δ [9]. The Hadamard Z2 Z4 -linear codes H are the Z2 Z4 -dual of the extended 1-perfect Z2 Z4 -linear codes. The rank of Hadamard Z2 Z4 -linear codes was computed in [23] and the rank of extended 1-perfect Z2 Z4 -linear codes in [7]. Specifically, rank(H ) = γ + 2δ + γ + 2δ δ  2 if δ ≥ 2 if δ = 0, 1 and rank(C) = γ̄ + 2δ̄ + δ = β + δ̄ + γ̄ , where γ̄ = α − γ and δ̄ = β − δ by (4) taking γ = κ. Note that the rank of these two families of Z2 Z4 -linear codes satisfies the upper bound. Example 3 Let Q R M(r, m) be the class of Z4 -linear Reed-Muller  codes defined in [3]. These are Z2 Z4 -linear codes of type (0, 2m ; 0, δ; 0), where δ = ri=0 mi . An important property is that any Z4 -linear Kerdock-like code of binary length 4m is in the class Q R M(1, 2m −1) and any extended Z4 -linear Preparata-like code of binary length 4m is in the class Q R M(2m − 3, 2m − 1). The rank of any code C ∈ Q R M(r, m) is t   r     m m rank(C) = , + i i i=0 i=0 where t = min(2r, m), [3]. Hence, if 2r ≥ m, then rank(C) = δ + β, i.e., the maximum possible. A Z4 -linear Kerdock-like code K of binary length 4m ≥ 16 has rank(P) = 2m 2 + m + 1 and an extended Z4 -linear Preparata-like code P of binary length 4m ≥ 64 has rank(P) = 22m − 2m [8], attaining the upper bound of Proposition 1. The next point to be solved is how to construct Z2 Z4 -linear codes with any rank in the range of possibilities given by Proposition 1. Lemma 4 There exists a Z2 Z4 -additive code C of type (α, β; γ , δ; κ) if and only if α, β, γ , δ, κ ≥ 0, α + β > 0, 0 < δ + γ ≤ β + κ and κ ≤ min(α, γ ). (5) ⊔ ⊓ Proof Straightforward from Theorem 1. Theorem 3 Let α, β, γ , δ, κ be integers satisfying (5). Then, there exists a Z2 Z4 -linear code C of type (α, β; γ , δ; κ) with rank(C) = r if and only if    δ r ∈ γ + 2δ, . . . , min β + δ + κ, γ + 2δ + . 2 Proof Let C be a Z2 Z4 -additive code of type (α, β; γ , δ; κ) with generator matrix ⎞ ⎛ Iκ T ′ 0 0 0 G = ⎝ 0 0 2T1 2Iγ −κ 0 ⎠ , 0 S ′ Sr 0 Iδ where Sr is a matrix over Z4 of size δ × (β − (γ − κ) − δ), and let C = (C ) be its corγ responding Z2 Z4 -linear code. Let {u i }i=1 and {v j }δj=1 be the row vectors of order two and four in G , respectively. 123 50 C. Fernández-Córdoba et al. The necessary conditions for the values 1, that is, rank(C) =  of r are given in Proposition   . In the generator matrix G , r = γ + 2δ + r̄ , where r̄ ∈ 0, . . . , min β − (γ − κ) − δ, 2δ γ the Gray map image of the γ row vectors {u i }i=1 and the 2δ row vectors {v j }δj=1 , {2v j }δj=1     , are independent binary vectors over Z2 . For each r̄ ∈ 0, . . . , min β − (γ − κ) − δ, 2δ we will define Sr in an appropriate way such that rank(C) = r = γ + 2δ + r̄ . Let ek , 1 ≤ k ≤ δ, denote the column vector of in the kth coordinate   length δ, with a one   , we can construct and zeroes elsewhere. For each r̄ ∈ 0, . . . , min β − (γ − κ) − δ, 2δ Sr as a quaternary matrix where in r̄ columns there are r̄ different column vectors ek + el of length δ, 1 ≤ k < l ≤ δ, and in the remaining columns there is the all-zero column vector. For each one of the r̄ column vectors the rank increases by 1. In fact, if the column vector ek + el is included in Sr , then the quaternary vector 2vk ∗ vl has only a two in the same coordinate where the column vector ek + el is and (2vk ∗ vl ) is independent to the vectors γ {(u i )}i=1 ,{(v j )}δj=1 , {(2v j )}δj=1 and {(2vs ∗ vt )}, {s, t}  = {k, l}. Since the maximum number of columns of Sr is β − (γ − κ) − δ and the maximum number of different such  columns is 2δ , the result follows. ⊔ ⊓ Note that in the proof of the previous theorem the matrices T ′ , T and S ′ can be chosen arbitrarily. Example 4 By Theorem 3, we know that the possible ranks for Z2 Z4 -linear codes of type (α, 9; 2, 5; 1) are r ∈ {12, 13, 14, 15}. For each possible r , we can construct a Z2 Z4 -linear code C with rank(C) = r , taking the following generator matrix of C = −1 (C): ⎞ 1 T′ 0 0 0 G S = ⎝ 0 0 2T1 2 0 ⎠ , 0 S ′ Sr 0 I5 ⎛ where S12 = (0) and S13 , S14 , and S15 are constructed as follows: ⎞ ⎛ ⎞ ⎛ ⎛ 1 1 0 0 1 0 0 ⎜1 ⎜1 1 0⎟ ⎜1 0 0⎟ ⎟ ⎟ ⎜ ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ S13 = ⎜ ⎜ 0 0 0 ⎟ , S14 = ⎜ 0 1 0 ⎟ , S15 = ⎜ 0 ⎝0 ⎝0 0 0⎠ ⎝0 0 0⎠ 0 0 0 0 0 0 0 0 1 1 0 0 ⎞ 1 0⎟ ⎟ 1⎟ ⎟. 0⎠ 0 4 Kernel dimension of Z2 Z4 -additive codes In this section, we will study the dimension of the kernel of Z2 Z4 -linear codes C = (C ). We will also show that there exists a Z2 Z4 -linear code C of type (α, β; γ , δ; κ) with k = ker (C) for any possible value of k. Lemma 5 Let C be a Z2 Z4 -additive code and let C = (C ) be the corresponding Z2 Z4 -linear code. Then, K (C) = {(u) | u ∈ C and 2u ∗ v ∈ C , ∀v ∈ C }. Proof By Lemma 2, (u) + (v) ∈ C if and only if 2u ∗ v ∈ C for all u, v ∈ C . Thus, the result follows. ⊔ ⊓ 123 Z2 Z4 -linear codes: rank and kernel 51 Note that if G is a generator matrix of a Z2 Z4 -additive code C and C = (C ), (u) ∈ K (C) if and only if u ∈ C and 2u ∗ v ∈ C for all row vector v in G . Moreover, all codewords of order two in C belong to K (C). Lemma 6 Let C be a Z2 Z4 -additive code and let C = (C ) be the corresponding Z2 Z4 -linear code. Given x, y ∈ C , (x) + (y) ∈ K (C) if and only if (x + y) ∈ K (C). Proof By Lemma 1, (x +y+2x ∗y) = (x)+(y). Now, by Lemma 5, (x +y+2x ∗y) ∈ K (C) if and only if for all v ∈ C , 2(x + y + 2x ∗ y) ∗ v = 2(x + y) ∗ v ∈ C ; that is, if and only if (x + y) ∈ K (C). ⊔ ⊓ Lemma 7 Let C be a Z2 Z4 -linear code of binary length n = α +2β and type (α, β; γ , δ; κ). Then, ker (C) ∈ {γ + δ, γ + δ + 1, . . . , γ + 2δ − 2, γ + 2δ}. Proof The upper bound γ +2δ comes from the linear case. The lower bound γ +δ is straightforward, since there are 2γ +δ codewords of order two in C = −1 (C) and, by Lemma 5, the binary images by  of all these codewords are in K (C). Also note that if the Z2 Z4 -linear code C is not linear, then the dimension of the kernel is equal to or less than γ + 2δ − 2 [21]. Therefore, ker (C) ∈ {γ + δ, . . . , γ + 2δ − 2, γ + 2δ}. ⊔ ⊓ β Given an integer m > 0, a set of vectors {v1 , v2 , . . . , vm } in Zα2 × Z4 and a subset I = {i 1 , . . . , il } ⊆ {1, . . . , m}, we denote by v I the vector vi1 +· · ·+vil . If I = ∅, then v I = 0. Note that given I, J ⊆ {1, . . . , m}, v I +v J = v I △J +2v I ∩J , where I △ J = (I ∪ J )−(I ∩ J ). Proposition 2 Let C be a Z2 Z4 -additive code of type (α, β; γ , δ; κ), with generator matrix G , and let C = (C ) be the corresponding Z2 Z4 -linear code with ker (C) = γ + 2δ − k̄, where k̄ ∈ {2, . . . , δ}. Then, there exists a set {v1 , v2 , . . . , vk̄ } of row vectors of order four in G , such that  (K (C) + (v I )) C= I ⊆{1,...,k̄} Proof We know that C can be written as the union of cosets of K (C) [1]. Since |K (C)| = 2γ +2δ−k̄ and |C| = 2γ +2δ , there are exactly 2k̄ cosets. Let u 1 , . . . , u γ , v1 , . . . , vδ be the γ and δ row vectors in G of order two and four, respectively. By Lemma 5, the binary images by  of all codewords of order two are in K (C). There are 2γ +δ codewords of order two generated by γ + δ codewords. Moreover, there are δ − k̄ codewords wi of order four such that (wi ) ∈ K (C) for all i ∈ {1, . . . , δ − k̄}, and (u 1 ), . . . , (u γ ), (2v1 ), . . . , (2vδ ), (w1 ), . . . , (wδ−k̄ ) are linearly independent vectors over Z2 . The code C can also be generated by u 1 , . . . , u γ , w1 , . . . , wδ−k̄ , vi1 , . . . , vik̄ , where {i 1 , i 2 , . . . , i k̄ } ⊆ {1, . . . , δ}. We can assume that vi1 , . . . , vik̄ are the k̄ row vectors v1 , . . . , vk̄ in G . Note that (v I )  ∈ K (C), for any I ⊆ {1, . . . , k̄} such that I  = ∅. In fact, if (v I ) ∈ K (C), then the set of vectors (u 1 ), . . . , (u γ ), (2v1 ), . . . , (2vδ ), (w1 ), . . . , (wδ−k̄ ), (v I ) would be linearly independent. Finally, we show that the 2k̄ − 1 binary vectors (v I ), I ⊆ {1, . . . , k̄} and I  = ∅, are in different cosets. Let (v I ) and (v J ) be any two of these binary vectors such that I  = J . If (v I ) ∈ K (C) + (v J ), then (v I ) + (v J ) ∈ K (C) and, by Lemma 6, (v I +v J ) ∈ K (C). We also have that v I +v J = v I △J +2v I ∩J . Hence, (v I △J +2v I ∩J ) = (v I △J ) + (2v I ∩J ) ∈ K (C) and (v I △J ) ∈ K (C), which is a contradiction, since I △ J ⊆ {1, . . . , k̄} and I △ J  = ∅. ⊔ ⊓ 123 52 C. Fernández-Córdoba et al. It is important to note that if C is a Z2 Z4 -linear code, then K (C) is a Z2 Z4 -linear subcode of C, by Lemma 6. The kernel of a Z2 Z4 -additive code C of type (α, β; γ , δ; κ), denoted by K(C ), can be defined as K(C ) = −1 (K (C)), where C = (C ) is the corresponding Z2 Z4 -linear code. By Lemma 5, K(C ) = {u ∈ C | 2u ∗ v ∈ C , ∀v ∈ C } and it is easy to see that K(C ) is a Z2 Z4 -additive subcode of C of type (α, β; γ + k̄, δ − k̄; κ). The following theorem follows. Theorem 4 Let C be a Z2 Z4 -linear code of type (α, β; γ , δ; κ) with ker (C) = γ + 2δ − k̄, where k̄ ∈ {0} ∪ {2, . . . , δ}. Then, K (C) is a Z2 Z4 -linear subcode of C of type (α, β; γ + k̄, δ − k̄; κ). Note that replacing ones with twos in the first α coordinates, we can see Z2 Z4 additive codes as quaternary linear codes. Let χ be the map from Z2 to Z4 , which is the usual inclusion from the additive structure in Z2 to Z4 : χ(0) = 0, χ(1) = 2. This map can be β α+β extended to the map (χ, I d) : Zα2 × Z4 → Z4 , which will also be denoted by χ. If C is a Z2 Z4 -additive code of type (α, β; γ , δ; κ) with generator matrix G , then χ(C ) is a quaternary linear code of length α + β and type 2γ 4δ with generator matrix Gχ (C ) = χ(G ). Note that K(C ) = χ −1 K(χ(C )) and K(χ(C ))⊥ is the quaternary linear code generated by the matrix   Hχ (C ) , 2Gχ (C ) ∗ Hχ (C ) where Hχ (C ) is the generator matrix of the quaternary dual code of χ(C ) and 2Gχ (C ) ∗ Hχ (C ) is the matrix obtained computing the component-wise product 2u ∗ v for all u ∈ Gχ (C ) , v ∈ Hχ (C ) . Moreover, by Proposition 2, given a Z2 Z4 -additive code C with generator matrix G , there exists a set {v1 , v2 , . . . , vk̄ } of row vectors of order four in G , such that  (K(C ) + v I ). C= I ⊆{1,...,k̄} Lemma 8 Let A be a symmetric matrix over Z2 of odd order and with zeroes in the main diagonal. Then, det(A) = 0. Proof Let n be the order of the matrix A. The map f : Zn2 × Zn2 → Zn2 defined by f (u, v) = u Av t is an alternating bilinear form and A is a symplectic matrix [19, 435]. It is known that the rank r of a symplectic matrix is always even [19, 436]. Therefore, since the order n of A is an odd number, r < n and det(A) = 0. ⊔ ⊓ Proposition 3 Let C be a Z2 Z4 -linear code of binary (α, β; γ , δ; κ). Then, ker (C) = γ + 2δ − k̄, where ⎧ ⎨ k̄ = 0, k̄ ∈ {0} ∪ {2, . . . , δ} and k̄ even, ⎩ k̄ ∈ {0} ∪ {2, . . . , δ}, length n = α + 2β and type i f s = 0, i f s = 1, i f s ≥ 2, and s = β − (γ − κ) − δ. Proof For s = 0, by Proposition 1 we have that rank(C) = γ + 2δ, so C is a binary linear code and ker (C) = γ + 2δ. For s ≥ 2, by Lemma 7 we have that ker (C) ∈ {γ + δ, . . . , γ + 2δ − 2, γ + 2δ}. 123 Z2 Z4 -linear codes: rank and kernel 53 Now, we will prove the result for s = 1. By Theorem 1, C is permutation equivalent to a Z2 Z4 -additive code generated by ⎞ Iκ T ′ 2T2 0 0 G S = ⎝ 0 0 2T1 2Iγ −κ 0 ⎠ , 0 S′ S R Iδ ⎛ γ where S is a matrix over Z4 of size δ × 1. Let {u i }i=1 and {v j }δj=1 be the row vectors in G S of order two and four, respectively. If δ < 3, then it is easy to see that ker (C) = γ + 2δ − 2 or ker (C) = γ + 2δ, by Lemma 7. If δ ≥ 3 we will show that, given four vectors v j1 , v j2 , v j3 , v j4 such that 2v j1 ∗ v j2  ∈ C and 2v j3 ∗ v j4  ∈ C , then 2v j1 ∗ v j2 + 2v j3 ∗ v j4 ∈ C . Let ek , 1 ≤ k ≤ γ − κ, denote the row vector of length γ − κ, with a one in the kth coordinate and zeroes elsewhere. Then, we can write 2v j1 ∗ v j2 = (0, 0, 2c, 2e I , 0), where c ∈ {0, 1} and I ⊆ {α + 2, . . . , α + γ − κ + 1}, and 2v j3 ∗ v j4 = (0, 0, 2c′ , 2e J , 0), where c′ ∈ {0, 1} and J ⊆ {α + 2, . . . , α + γ − κ + 1}. We denote by u I (resp. u J ) the row vector obtained by adding the row vectors of order two in G S with 2 in the coordinate positions given by I (resp. J ). Then, u I = (0, 0, 2d, 2e I , 0) ∈ C with d ∈ {0, 1} (resp. u J = (0, 0, 2d ′ , 2e J , 0) ∈ C with d ′ ∈ {0, 1}). Since 2v j1 ∗ v j2  ∈ C (resp. 2v j3 ∗v j4  ∈ C ) we have 2v j1 ∗v j2 = u I +(0, 0, 2, 0, 0) (resp. 2v j3 ∗v j4 = u J +(0, 0, 2, 0, 0)). Therefore, 2v j1 ∗ v j2 + 2v j3 ∗ v j4 = u I + u J ∈ C By Proposition 2, there exist k̄ row vectors v1 , v2 , . . . , vk̄ in G S , such that (v I )  ∈ K (C) for any nonempty subset I ⊆ {1, . . . , k̄} and ker (C) = γ + 2δ − k̄. Assume k̄ is odd. We will show that there exists a subset I ⊆ {1, . . . , k̄} such that (v I ) ∈ K (C). Since this is a contradiction, k̄ can not be an odd number and the assertion will be proved. By Lemma 5, in order to prove that (v I ) ∈ K (C), it is enough to prove that 2v I ∗ v j ∈ C for all j ∈ {1, . . . , k̄}. That is, following the above remark, for each j ∈ {1, . . . , k̄} the number of i ∈ I such that 2vi ∗ v j  ∈ C is even. We define a symmetric matrix A = (ai j ), 1 ≤ i, j ≤ k̄, in the following way: ai j = 1 if 2vi ∗ v j  ∈ C and 0 otherwise. Therefore, A is a symmetric matrix of odd order and with zeroes in the main diagonal. Lemma 8 shows that det(A) = 0 and hence there exists a linear combination of some rows, i 1 , . . . , il , of A equal to 0. The vector (v I ), where I = {i 1 , . . . , il }, belongs to K (C). This completes the proof. ⊔ ⊓ Example 5 Continuing with Example 1, the dimension of the kernel for a Hadamard Z4 -linear code H was computed in [23,17] and the dimension of the kernel for an extended 1-perfect Z4 -linear code C in [7]. Specifically, ker (H ) = γ + δ + 1 if δ ≥ 3 γ + 2δ if δ = 1, 2 and ⎧ ⎨ γ̄ + δ̄ + 1 if δ ≥ 3 ker (C) = γ̄ + δ̄ + 2 if δ = 2 ⎩ γ̄ + δ̄ + t if δ = 1. Example 6 Continuing with Example 2, the dimension of the kernel for a Hadamard Z2 Z4 linear code H was computed in [23] and the dimension of the kernel for an extended 1-perfect Z2 Z4 -linear code C in [7]. Specifically, ker (H ) = γ + δ if δ ≥ 2 γ + 2δ if δ = 0, 1 123 54 C. Fernández-Córdoba et al. and ker (C) = γ̄ + δ̄ + 1 if δ ≥ 1 γ̄ + 2δ̄ if δ = 0. Note that the kernel dimension of the Hadamard Z2 Z4 -linear codes satisfies the lower bound. Example 7 Let Q R M(r, m) be the class of Z4 -linear Reed-Muller codes defined in [3], as in Example 3. The dimension of the kernel of any code C ∈ Q R M(r, m) is r    m + 1 = δ + 1, ker (C) = i i=0 2m+1 except for r = m (in this case, C = Z2 ), [3]. Therefore, Z4 -linear Kerdock-like codes and extended Z4 -linear Preparata-like codes of binary length 4m have dimension of the kernel ker (K ) = 2m + 1 and ker (P) = 22m−1 − 2m + 1, respectively [3,8]. As in Sect. 3 for the rank, the next point to be solved here is how to construct Z2 Z4 -linear codes with any dimension of the kernel in the range of possibilities given by Proposition 3. Theorem 5 Let α, β, γ , δ, κ be integers satisfying (5). Then, there exists a Z2 Z4 -linear code C of type (α, β; γ , δ; κ) with ker (C) = γ + 2δ − k̄ if and only if ⎧ if s = 0, ⎨ k̄ = 0, k̄ ∈ {0} ∪ {2, . . . , δ} and k̄ even, if s = 1, ⎩ k̄ ∈ {0} ∪ {2, . . . , δ}, if s ≥ 2, and s = β − (γ − κ) − δ. Proof Let C be a Z2 Z4 -additive code of type (α, β; γ , δ; κ) with generator matrix ⎞ ⎛ 0 0 Iκ T ′ 0 G = ⎝ 0 0 0 2Iγ −κ 0 ⎠ , 0 S ′ Sk 0 Iδ γ where Sk is a matrix over Z4 of size δ × s, and {u i }i=1 and {v j }δj=1 are the row vectors in G of order two and four, respectively. Let C = (C ) be the corresponding Z2 Z4 -linear code with ker (C) = k = γ + 2δ − k̄. The necessary conditions for the values of k are given in Proposition 3. When s = 0, the code C is a binary linear code, so ker (C) = k = γ + 2δ and k̄ = 0. If s > 0, then taking Sk the all-zero matrix over Z4 , the code C is also a binary linear code and k̄ = 0. When s = 1, for each k̄ ∈ {2, . . . , δ} and even, we can construct a matrix Sk over Z4 of size δ × 1 with an even number of ones, k̄, and zeroes elsewhere. We can assume that v1 , . . . , vk̄ are the row vectors of order four with 1 in the column from the matrix Sk , and vk̄+1 , . . . , vδ the ones with 0 in this column. By Lemma 5, it is easy to see that (v j ) ∈ K (C) for all j ∈ {k̄ + 1, . . . , δ}. Since the binary images by  of all codewords of order two are also in K (C), ker (C) ≥ γ + δ + δ − k̄ = γ + 2δ − k̄. Let K ′ (C) be the binary linear subcode of K (C) generated by these γ+ 2δ − k̄ linearly independent vectors. By the same arguments as in Proposition 2, C = I ⊆{1,...,k̄} (K ′ (C) + (v I )). In order to prove that ker (C) = γ + 2δ − k̄, it is enough to show that a codeword from each coset does not belong 123 Z2 Z4 -linear codes: rank and kernel 55 to K (C), that is, (v I )  ∈ K (C) for any nonempty subset I ⊆ {1, . . . , k̄}. If |I | is even, 2v I ∗ v j  ∈ C for any j ∈ I . If |I | is odd, 2v I ∗ v j  ∈ C for any j ∈ {1, . . . , k̄}\I . Note that {1, . . . , k̄}\I is a nonempty set, since k̄ ≥ 2 and even. Therefore, (v I )  ∈ K (C) by Lemma 5. Finally, when s ≥ 2, for each k̄ ∈ {2, 3, . . . , δ}, we can construct a matrix Sk over Z4 of size δ × s, such that only in the last δ − k̄ row vectors all components are zero and, moreover, in the first k̄ coordinates of each column vector there are an even number of ones and zeros elsewhere. In this case, by the same arguments as before, it is easy to prove that ker (C) = γ + 2δ − k̄. ⊔ ⊓ Example 8 By Theorem 5, we know that the possible dimensions of the kernel for Z2 Z4 linear codes of type (α, 9; 2, 5; 1) are k ∈ {12, 10, 9, 8, 7}. For each possible k, we can construct a Z2 Z4 -linear code C with ker (C) = k, taking the following generator matrix of C = −1 (C): ⎞ ⎛ 1 T′ 0 0 0 GS = ⎝ 0 0 0 2 0 ⎠ , 0 S ′ Sk 0 I5 where S12 = (0) and S10 , S9 , S8 and S7 ⎞ ⎛ ⎛ 1 0 0 1 0 ⎜1 0 0⎟ ⎜1 1 ⎟ ⎜ ⎜ ⎟ ⎜ S10 = ⎜ ⎜ 0 0 0 ⎟ , S9 = ⎜ 0 1 ⎝0 0 0⎠ ⎝0 0 0 0 0 0 0 are constructed as follows: ⎞ ⎞ ⎛ ⎛ 0 1 0 0 1 ⎜1 1 0⎟ ⎜1 0⎟ ⎟ ⎟ ⎜ ⎜ ⎟ ⎜ ⎜ 0⎟ ⎟ , S8 = ⎜ 1 1 0 ⎟ , S7 = ⎜ 1 ⎝1 0 0⎠ ⎝1 0⎠ 0 0 0 0 0 0 1 1 1 1 ⎞ 0 0⎟ ⎟ 0⎟ ⎟. 0⎠ 0 5 Pairs of rank and kernel dimension of Z2 Z4 -additive codes In this section, once the dimension of the kernel is fixed, lower and upper bounds on the rank are established. We will show that there exists a Z2 Z4 -linear code C of type (α, β; γ , δ; κ) with r = rank(C) and k = ker (C) for any possible pair of values (r, k). Lemma 9 Let C be a Z2 Z4 -additive code of type (α, β; γ , δ; κ) and let C = (C ) be the corresponding Z2 Z4 -linear code. If rank(C) = γ + 2δ + r̄ and ker (C) = γ + 2δ − k̄, with k̄ ≥ 2, then   k̄ 1 ≤ r̄ ≤ . 2 Moreover, C is generated by γ {(u i )}i=1 , {(v j ), (2v j )}δj=1 and {(2vt ∗ vs )}1≤s<t≤k̄ . γ Proof There exist {u i }i=1 and {v j }δj=1 vectors of order two and four respectively, such that  they generate the code C and C = I ⊆{1,...,k̄} (K (C) + (v I )) by Proposition 2. Note that (v j ) ∈ K (C) if and only if j ∈ {k̄ + 1, . . . , δ}. By Lemma 5, for all j ∈ {k̄ +1, . . . , δ} and i ∈ {1, . . . , δ}, as (v j ) ∈ K (C), 2v j ∗vi ∈ C γ and, consequently, (2v j ∗ vi ) is a linear combination of {(u i )}i=1 and {(2v j )}δj=1 . As γ a result, C is generated by {(u i )}i=1 , {(v j ), (2v j )}δj=1 and {(2vt ∗ vs )}1≤s<t≤k̄ and k̄  hence r̄ ≤ 2 , by Lemma 3. Finally, since k̄ ≥ 2, the binary code C is not linear and, therefore, r̄ ≥ 1. ⊔ ⊓ 123 56 C. Fernández-Córdoba et al. Let C be a Z2 Z4 -linear code with ker (C) = γ + 2δ − k̄ and rank(C) = γ + 2δ + r̄ . Note that if r̄ = 0 then, necessarily, k̄ = 0 (and vice versa) and C is a linear code. The next theorem will determine all possible pairs of rank and dimension of the kernel for nonlinear Z2 Z4 -linear codes. Proposition 4 Let C be a nonlinear Z2 Z4 -linear code of binary length n = α + 2β and type (α, β; γ , δ; κ) with ker (C) = γ + 2δ − k̄ and rank(C) = γ + 2δ + r̄ . Then, for any k̄ ∈ {2, . . . , δ},   r̄ ∈ {2, . . . , min(β − (γ − κ) − δ, k̄2 )}, if k̄ is odd,  r̄ ∈ {1, . . . , min(β − (γ − κ) − δ, k̄2 )}, if k̄ is even.  Proof By Proposition 1, r̄ ∈ {0, . . . , min(β − (γ − κ) − δ, 2δ )}. Moreover, by Lemma 9, k̄  for a fixed k̄ ≥ 2, r̄ ≤ 2 and, therefore, if k̄ ∈ {2, . . . , δ} then r̄ ∈ {1, . . . , min(β − (γ −  κ) − δ, k̄2 )}. In the case r̄ = 1, C is not linear and, by Lemma 7, ker (C) = γ + 2δ − k̄ where k̄ ∈ {2, . . . , δ}. Moreover, by Proposition 2, there exists a generator matrix G of C with row γ vectors {u i }i=1 and {v j }δj=1 of order two and four, respectively, such that the k̄ row vectors  v1 , v2 , . . . , vk̄ satisfy C = I ⊆{1,...,k̄} (K (C) + (v I )). We will see that if r̄ = 1, then k̄ is necessarily even. Assume k̄ is odd. We will prove that there exist I ⊆ {1, . . . , k̄} such that (v I ) ∈ K (C), that is, 2v I ∗ v j ∈ C for all j ∈ {1, . . . k̄}, which is a contradiction and, therefore, k̄ is an even number. γ By Lemma 9, C is generated by {(u i )}i=1 , {(v j ), (2v j )}δj=1 and {(2v j ∗ vk )}1≤ j<k≤k̄ . Since rank(C) = γ + 2δ + 1, for all i, j ∈ {1, . . . , k̄} either 2vi ∗ v j ∈ C or 2vi ∗ v j = w, for an order two vector w  ∈ C . If there exist I ⊆ {1, . . . , k̄} such that, for each j ∈ {1, . . . , k̄} the number of i ∈ I verifying 2vi ∗ v j = w ∈ / C is even, then 2v I ∗ v j ∈ C . In order to prove that there exists such a set I , we define the symmetric matrix A = (ai j ), 1 ≤ i, j ≤ k̄, as in the proof of Proposition 3, and we get the contradiction. ⊔ ⊓ Theorem 6 Let α, β, γ , δ, κ be integers satisfying (5). Then, there exists a nonlinear Z2 Z4 linear code C of type (α, β; γ , δ; κ) with ker (C) = γ + 2δ − k̄ and rank(C) = γ + 2δ + r̄ if and only k̄ ∈ {2, . . . , δ} and   ⎧   ⎨ r̄ ∈ 2, . . . , min β − (γ − κ) − δ, k̄2 , if k̄ is odd,      k̄ ⎩ r̄ ∈ 1, . . . , min β − (γ − κ) − δ, , if k̄ is even. 2 Proof Let C be a Z2 Z4 -additive code of type (α, β; γ , δ; κ) with generator matrix ⎞ ⎛ Iκ T ′ 0 0 0 G = ⎝ 0 0 0 2Iγ −κ 0 ⎠ , 0 S ′ Sr,k 0 Iδ where Sr,k is a matrix over Z4 of size δ × (β − (γ − κ) − δ), and let C = (C ) be its corresponding Z2 Z4 -linear code. The necessary conditions for the values of the pairs r̄ and k̄ are given in Proposition 4. and Let ek , 1 ≤ k ≤ δ, denote the column vector of length  δ, with a one  in the kth coordinate   zeroes elsewhere. For each k̄ ∈ {3, . . . , δ} and r̄ ∈ 2, . . . , min β − (γ − κ) − δ, k̄2 , 123 Z2 Z4 -linear codes: rank and kernel 57 we can construct Sr,k as a quaternary matrix where in one column there is the vector e1 + · · · + ek̄ , in r̄ − 1 columns there are r̄ − 1 different column vectors ek + el of length δ, 1 ≤ k < l ≤ k̄, and in the remaining columns there is the all-zero column vector. It is easy to check that ker (C) = γ + 2δ − k̄ and rank(C) = γ + 2δ + r̄ . Finally, if r̄ = 1, we can construct Sr,k as a quaternary matrix of size δ × (β − (γ − κ) − δ) with k̄ ones in one column and zeroes elsewhere, for each k̄ ∈ {2, . . . , δ} and even. In this case, it is also easy to check that rank(C) = γ + 2δ + 1 and ker (C) = γ + 2δ − k̄, for any k̄ ∈ {2, . . . , δ} and even. ⊔ ⊓ Example 9 By Theorem 6, we know that the possible pairs of rank and dimension of the kernel of Z2 Z4 -linear codes, C, of type (α, 9; 2, 5; 1) are given in the following table: k \ r 12 13 14 15 12 ∗ 10 ∗ 9 ∗ ∗ 8 ∗ ∗ ∗ 7 ∗ ∗ For each possible pair (r, k), we can construct a Z2 Z4 -linear code C with rank(C) = r and ker (C) = k, taking the following generator matrix of C = −1 (C): ⎛ ⎞ 1 T′ 0 0 0 GS = ⎝ 0 0 0 2 0 ⎠ , 0 S ′ Sr,k 0 I5 where S12,12 = (0) and the other possible Sr,k are constructed as follows: ⎞ ⎛ ⎛ 1 0 0 1 0 ⎜1 0 0⎟ ⎜1 0 ⎟ ⎜ ⎜ ⎟ ⎜ S13,10 = ⎜ ⎜ 0 0 0 ⎟ , S13,8 = ⎜ 1 0 ⎝0 0 0⎠ ⎝1 0 0 0 0 0 0 ⎞ ⎞ ⎛ ⎛ ⎛ 1 1 1 1 0 1 1 0 ⎜1 1 ⎜1 1 0⎟ ⎜1 1 0⎟ ⎟ ⎟ ⎜ ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ S14,9 = ⎜ ⎜ 1 0 0 ⎟ , S14,8 = ⎜ 1 0 0 ⎟ , S14,7 = ⎜ 1 0 ⎝1 0 ⎝1 0 0⎠ ⎝0 0 0⎠ 1 0 0 0 0 0 0 0 ⎞ ⎞ ⎛ ⎛ ⎛ 1 1 1 1 0 1 1 0 ⎜1 1 ⎜1 1 1⎟ ⎜1 1 1⎟ ⎟ ⎟ ⎜ ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ S15,9 = ⎜ ⎜ 1 0 1 ⎟ , S15,8 = ⎜ 1 0 1 ⎟ , S15,7 = ⎜ 1 0 ⎝1 0 ⎝1 0 0⎠ ⎝0 0 0⎠ 1 0 0 0 0 0 0 0 ⎞ 0 0⎟ ⎟ 0⎟ ⎟, 0⎠ 0 ⎞ 0 0⎟ ⎟ 0⎟ ⎟, 0⎠ 0 ⎞ 0 1⎟ ⎟ 1⎟ ⎟. 0⎠ 0 6 Conclusion In this paper we studied two structural properties of Z2 Z4 -linear codes, the rank and dimension of the kernel. Using combinatorial enumeration techniques, we established lower and upper bounds for the possible values of these parameters. We also gave the construction of a Z2 Z4 -linear code with rank r (resp. kernel dimension k) for each feasible value r (resp. k). 123 58 C. Fernández-Córdoba et al. Finally, we established the bounds on the rank, once the dimension of the kernel is fixed, and we gave the construction of a Z2 Z4 -linear code with rank r and kernel dimension k for each possible pair (r, k). The rank, kernel and dimension of the kernel are defined for binary codes and they are specially useful for binary nonlinear codes. We showed that for binary codes which are Z2 Z4 -linear codes, we can also define the kernel using the corresponding Z2 Z4 -additive β codes, which are subgroups of Zα2 × Z4 . In this case, in order to compute the kernel K (C) of a Z2 Z4 -linear code C is much easier if we consider the corresponding Z2 Z4 -additive code C = −1 (C) and we compute K(C ) = −1 (K (C)) using a generator matrix of C . Moreover, we also proved that if C is a Z2 Z4 -linear code, then K (C) and C are also Z2 Z4 -linear codes. Finally, since K (C) ⊆ C ⊆ C and C can be written as the union of cosets of K (C), we also have that, equivalently, K(C ) ⊆ C ⊆ SC , where SC = −1 (C), and C can be written as cosets of K(C ). As a future research in this issue, it would be interesting to establish a characterization of all Z2 Z4 -linear codes of type (α, β; γ , δ; κ) with rank r and dimension of the kernel k, using the canonical generator matrices G S of the form (3) and characterizing their submatrices Sr,k over Z4 of size δ × (β − γ − δ). Acknowledgments This work has been partially supported by the Spanish MICINN grant MTM2009-08435 and the Catalan AGAUR grant 2009SGR1224. First author wishes to acknowledge the joint sponsorship of the Fulbright Program in Spain and the Ministry of Science and Innovation during a research stay at Auburn University. References 1. Bauer H., Ganter B., Hergert F.: Algebraic techniques for nonlinear codes. Combinatorica 3, 21–33 (1983). 2. Bierbrauer J.: Introduction to coding theory. Chapman & Hall/CRC, Boca Raton (2005). 3. Borges J., Fernández C., Phelps K.T.: Quaternary Reed–Muller codes. IEEE Trans. Inform. Theory. 51(7), 2686–2691 (2005). 4. Borges J., Fernández-Córdoba C., Phelps K.T.: ZRM codes. IEEE Trans. Inform. Theory. 54(1), 380–386 (2008). 5. Borges J., Fernández C., Pujol J., Rifà J., Villanueva M.: Z2 Z4 -linear codes: generator matrices and duality, to appear in designs, codes and cryptography. arXiv:0710.1149 (2009). 6. Borges J., Fernández C., Pujol J., Rifà J., Villanueva M.: Z2 Z4 -additive codes. A Magma package. Universitat Autònoma de Barcelona. http://www.ccg.uab.cat (2007). 7. Borges J., Phelps K.T., Rifà J.: The rank and kernel of extended 1-perfect Z4 -linear and additive non-Z4 linear codes. IEEE Trans. Inform. Theory. 49(8), 2028–2034 (2003). 8. Borges J., Phelps K.T., Rifà J., Zinoviev V.A.: On Z4 -linear preparata-like and kerdock-like codes. IEEE Trans. Inform. Theory. 49(11), 2834–2843 (2003). 9. Borges J., Rifà J.: A characterization of 1-perfect additive codes. IEEE Trans. Inform. Theory. 45(5), 1688–1697 (1999). 10. Cannon J.J., Bosma W.: (eds.) Handbook of Magma functions, edn. 2.13. 4350 pp. University of Kashan, Kashan, Iran (2006). 11. Delsarte P.: An algebraic approach to the association schemes of coding theory. Philips Res. Rep. Suppl. 10(2), 1–97 (1973). 12. Delsarte P., Levenshtein V.: Asociation schemes and coding theory. IEEE Trans. Inform. Theory. 44(6), 2477–2504 (1998). 13. Fernández C., Villanueva M.: Z2 Z4 -linear codes with all possible ranks and kernel dimensions, in XI international symposium on problems of redundancy in information and control systems. St. Petersburg, Russia, pp. 62–65, July (2007). 14. Fernández-Córdoba C., Pujol J., Villanueva M.: On rank and kernel of Z4 -linear codes. Lecture Notes in Computer Science no. 5228, pp. 46–55 (2008). 15. Hammons A.R., Kumar P.V., Calderbank A.R., Sloane N.J.A., Solé P.: The Z4 -linearity of kerdock, preparata, goethals and related codes. IEEE Trans. Inform. Theory. 40, 301–319 (1994). 123 Z2 Z4 -linear codes: rank and kernel 59 16. Huffman W.C., Pless V.: Fundamentals of error-correcting codes. Cambridge University Press, Cambridge (2003). 17. Krotov D.S.: Z4 -linear Hadamard and extended perfect codes. Electron. Note Discr. Math. 6, 107–112 (2001). 18. Lindström B.: Group partitions and mixed perfect codes. Canad. Math. Bull. 18, 57–60 (1975). 19. MacWillams F.J., Sloane N.J.A.: The theory of error correcting codes. North Holland, Amsterdam (1977). 20. Pernas J., Pujol J., Villanueva M.: Kernel dimension for some families of quaternary Reed–Muller codes. Lecture Notes in Computer Science n. 5393. pp. 128–141 (2008). 21. Phelps K.T., LeVan M.: Kernels of nonlinear Hamming codes. Design code cryptogr. 6(3), 247–257 (1995). 22. Phelps K.T., Rifà J.: On binary 1-perfect additive codes: some structural properties. IEEE Trans. Inform. Theory. 48(9), 2587–2592 (2002). 23. Phelps K.T., Rifà J., Villanueva M.: On the additive Z4 -linear and non-Z4 -linear Hadamard codes. Rank and Kernel. IEEE Trans. Inform. Theory. 52(1), 316–319 (2005). 24. Pujol J., Rifà J.: Translation invariant propelinear codes. IEEE Trans. Inform. Theory. 43, 590–598 (1997). 25. Pujol J., Rifà J., Solov’eva F.I.: Construction of Z4 -linear Reed–Muller codes. IEEE Trans. Inform. Theory. 55(1), 99–104 (2009). 26. Wan Z.-X.: Quaternary codes. World Scientific, Singapore (1997). 123