Abstract The $$\mathbb {Z}_2\mathbb {Z}_4\mathbb {Z}_8$$ Z 2 Z 4 Z 8 -additive codes are subgroups of $$\mathbb {Z}_2^{\alpha _1} \times \mathbb {Z}_4^{\alpha _2} \times \mathbb {Z}_8^{\alpha _3}$$ Z 2 α 1 × Z 4 α 2 × Z 8 α 3 . A $$\mathbb {Z}_2\mathbb {Z}_4\mathbb {Z}_8$$ Z 2 Z 4 Z 8 -linear Hadamard code is a Hadamard code which is the Gray map image of a $$\mathbb {Z}_2\mathbb {Z}_4\mathbb {Z}_8$$ Z 2 Z 4 Z 8 -additive code. A recursive construction of $$\mathbb {Z}_2\mathbb {Z}_4\mathbb {Z}_8$$ Z 2 Z 4 Z 8 -additive Hadamard codes of type $$(\alpha _1,\alpha _2, \alpha _3;t_1,t_2, t_3)$$ ( α 1 , α 2 , α 3 ; t 1 , t 2 , t 3 ) with $$\alpha _1 \ne 0$$ α 1 ≠ 0 , $$\alpha _2 \ne 0$$ α 2 ≠ 0 , $$\alpha _3 \ne 0$$ α 3 ≠ 0 , $$t_1\ge 1$$ t 1 ≥ 1 , $$t_2 \ge 0$$ t 2 ≥ 0 , and $$t_3\ge 1$$ t 3 ≥ 1 is known. In this paper, we generalize some known results for

The Zps-additive codes of lengthnare subgroups of Znps, withpprime ands≥ 1. They can be seen as a generalization of linear codes over Z2, Z4, or more general over Z2s. In this paper, we show two methods for computing a parity-check matrix of a Zps-additive code from a generator matrix of the code in standard form. We also compare the performance of our results implemented in Magma with the current available function in Magma for linear codes over finite rings in general. Complementing this comparison, we also show a time complexity analysis of the algorithms. The rings Zpsbelong to a more general class of rings: finite chain rings. Along the paper, we observe that the same results can be applied to any linear code over a finite commutative chain ring.

The$\mathbb{Z}_{2}\mathbb{Z}_{4}\mathbb{Z}_{8}$-additive codes are subgroups of$\mathbb{Z}_{2}^{\alpha_{1}}\times \mathbb{Z}_{4}^{\alpha_{2}}\times \mathbb{Z}_{8}^{\alpha_{3}}$. A$\mathbb{Z}_{2}\mathbb{Z}_{4}\mathbb{Z}_{8}$-linear Hadamard code is a Hadamard code which is the Gray map image of a$\mathbb{Z}_{2}\mathbb{Z}_{4}\mathbb{Z}_{8}$-additive code. A recursive construction of$\mathbb{Z}_{2}\mathbb{Z}_{4}\mathbb{Z}_{8}$-additive Hadamard codes of type$(\alpha_{1}, \alpha_{2}, \alpha_{3};t_{1}, t_{2},t_{3})$with$\alpha_{1}\neq 0, \alpha_{2}\neq 0, \alpha_{3}\neq 0,t_{1}\geq 1, t_{2}\geq 0$, and$t_{3} > 1$is known, and for which types the corresponding$\mathbb{Z}_{2}\mathbb{Z}_{4}\mathbb{Z}_{8}$-linear Hadamard codes are binary nonlinear codes is also known. In this paper, we generalize some known results for$\mathbb{Z}_{2}\mathbb{Z}_{4}$-linear Hadamard codes to$\mathbb{Z}_{2}\mathbb{Z}_{4}\mathbb{Z}_{8}$-linear Hadamard codes with$\alpha_{1}\neq 0, \alpha_{2}\neq 0$, and$\alpha_{3}\neq 0$. First, for these codes, we compute the kernel and its dimension whenever they are nonlinear, which allows us to give a partial classification of these codes. Moreover, for$3\leq t\leq 11$, we give a complete classification by providing the exact amount of nonequivalent such codes of length$2^{t}$. We also give several families of infinite such nonlinear$\mathbb{Z}_{2}\mathbb{Z}_{4}\mathbb{Z}_{8}$-linear Hadamard codes, which are not equivalent to any other constructed$\mathbb{Z}_{2}\mathbb{Z}_{4}\mathbb{Z}_{8}$-linear Hadamard code, nor to any$\mathbb{Z}_{2}\mathbb{Z}_{4}$-linear Hadamard code, nor to any previously constructed$\mathbb{Z}_{2^{s}}$-linear Hadamard code with$s\geq 2$, with the same length$2^{t}$.

The$\mathbb{Z}_{p^{s}}-\mathbf{additive}$codes of length$n$are subgroups of$\mathbb{Z}_{p^{s}}^{n}$, with$p$prime and$s\geq 1$. They can be seen as a generalization of linear codes over$\mathbb{Z}_{2},\ \mathbb{Z}_{4}$, or more general over$\mathbb{Z}_{2^{s}}$. In this paper, we show two methods for computing a parity-check matrix of a$\mathbb{Z}_{p^{s}} -\mathbf{additive}$code from a generator matrix of the code in standard form. We also compare the performance of our results implemented in Magma with the current available function in Magma for codes over finite rings in general.

Abstract Linear codes of length n over $$\mathbb {Z}_{p^s}$$ Z p s , p prime, called $$\mathbb {Z}_{p^s}$$ Z p s -additive codes, can be seen as subgroups of $$\mathbb {Z}_{p^s}^n$$ Z p s n . A $$\mathbb {Z}_{p^s}$$ Z p s -linear generalized Hadamard (GH) code is a GH code over $$\mathbb {Z}_p$$ Z p which is the image of a $$\mathbb {Z}_{p^s}$$ Z p s -additive code under a generalized Gray map. It is known that the dimension of the kernel allows to classify these codes partially and to establish some lower and upper bounds on the number of such codes. Indeed, in this paper, for $$p\ge 3$$ p ≥ 3 prime, we establish that some $$\mathbb {Z}_{p^s}$$ Z p s -linear GH codes of length $$p^t$$ p t having the same dimension of the kernel are equivalent to each other, once t is fixed. This allows us to improve the known upper bounds. Moreover, up to $$t=10$$ t = 10 if $$p=3$$ p = 3 or $$t=8$$ t = 8 if $$p=5$$ p = 5 , this new upper bound coincides with a known lower bound based on the rank and dimension of the kernel.

It is known that$\mathbb {Z}_{p^{s}}$-linear codes, which are the Gray map image of$\mathbb {Z}_{p^{s}}$-additive codes (linear codes over$\mathbb {Z}_{p^{s}}$), are systematic and a systematic encoding has been found. This makes$\mathbb {Z}_{p^{s}}$-linear codes suitable to apply the permutation decoding method, based on the existence of r-PD-sets, which are subsets of the permutation automorphism group of the code. Some constructions of r-PD-sets of minimum size$r+1$for$\mathbb {Z}_{p^{s}}$-linear generalized Hadamard codes of type$(n;t_{1}, {\dots },t_{s})$are known. In this paper, for these codes, we present new constructions of r-PD-sets of size$r+1$, which are suitable for all parameters$t_{1}, {\dots },t_{s}$. These allow us to obtain new r-PD-sets for values of r closer to the theoretical upper bound, improving previous known results.

The ℤ2ℤ4ℤ8-additive codes are subgroups of $\mathbb{Z}_2^{{\alpha _1}} \times \mathbb{Z}_4^{{\alpha _2}} \times \mathbb{Z}_8^{{\alpha _3}}$. A ℤ2ℤ4ℤ8-linear Hadamard code is a Hadamard code which is the Gray map image of a ℤ2ℤ4ℤ8-additive code. In this paper, we generalize some known results for ${\mathbb{Z}_2}{\mathbb{Z}_4}$-linear Hadamard codes to ℤ2ℤ4ℤ8-linear Hadamard codes with ${\alpha _1} \ne 0$, ${\alpha _2} \ne 0$, and ${\alpha _3} \ne 0$. First, we give a recursive construction of ℤ2ℤ4ℤ8-additive Hadamard codes of type $\left( {{\alpha _1},{\alpha _2},{\alpha _3};{t_1},{t_2},{t_3}} \right)$ with ${t_1} \geq 1,{t_2} \geq 0$, and ${t_3} \geq 1$. Then, we show for which types the corresponding ℤ2ℤ4ℤ8-linear Hadamard codes are nonlinear over ${\mathbb{Z}_2}$. Moreover, we show that, unlike ${\mathbb{Z}_2}{\mathbb{Z}_4}$-linear Hadamard codes, in general, this family of ℤ2ℤ4ℤ8-linear Hadamard codes does not include the family of ℤ4-linear or ${\mathbb{Z}_8}$-linear Hadamard codes. Actually, we show that, for example, for length ${2^{11}}$, the constructed nonlinear ℤ2ℤ4ℤ8-linear Hadamard codes are not equivalent to each other, nor to any ${\mathbb{Z}_2}{\mathbb{Z}_4}$-linear Hadamard, nor to any previously constructed ${\mathbb{Z}_{{2^s}}}$-linear Hadamard code, with $s \geq 2$.

The ${\mathbb{Z}_p}{\mathbb{Z}_{{p^2}}}{\text{ - additive}}$ codes are subgroups of $\mathbb{Z}_p^{{\alpha _1}} \times \mathbb{Z}_{{p^2}}^{{\alpha _2}}$. A${\mathbb{Z}_p}{\mathbb{Z}_{{p^2}}}{\text{ - linear}}$ generalized Hadamard (GH) code is a GH code over ${\mathbb{Z}_p}$ which is the Gray map image of a ${\mathbb{Z}_p}{\mathbb{Z}_{{p^2}}}$-additive code. A recursive construction of ${\mathbb{Z}_p}{\mathbb{Z}_{{p^2}}}{\text{ - additive}}$ GH codes of type (α1, α2; t1, t2) with t1, t2≥ 1 is known, and for which types the corresponding ${\mathbb{Z}_p}{\mathbb{Z}_{{p^2}}}$-linear GH codes are nonlinear over ${\mathbb{Z}_p}$ is also known. In this paper, we generalize some known results for ${\mathbb{Z}_p}{\mathbb{Z}_{{p^2}}}$-linear GH codes with p = 2 to any p≥3 prime when ${\alpha _1} \ne 0$. First, we present new recursive constructions of ${\mathbb{Z}_p}{\mathbb{Z}_{{p^2}}}{\text{ - linear}}$ GH codes having the same type, and show that we obtained equivalent codes. Then, we compute the rank of some families of ${\mathbb{Z}_p}{\mathbb{Z}_{{p^2}}}{\text{ - Linear }}$ GH codes. Finally, we show that, unlike ${\mathbb{Z}_4}{\text{ - linear}}$ Hadamard codes, the ${\mathbb{Z}_{{p^2}}}{\text{ - Linear }}$ GH codes are not included in the family of ${\mathbb{Z}_p}{\mathbb{Z}_{{p^2}}}{\text{ - Linear }}$ GH codes with ${\alpha _1} \ne 0$ when p ≥ 3 prime.

In a previous work it was shown that ${\mathbb{Z}_{{p^s}}}$-linear codes, which are the Gray map image of ${\mathbb{Z}_{{p^s}}}$-additive codes (linear codes over ${\mathbb{Z}_{{p^s}}}$), are systematic by giving a systematic encoding. This makes ${\mathbb{Z}_{{p^s}}}$-linear codes and, in particular, ${\mathbb{Z}_8}$-linear codes suitable to apply the permutation decoding method. This technique is also based on the existence of s-PD-sets, which are subsets of the permutation automorphism group of the code. In this paper, we study the permutation automorphism group of ${\mathbb{Z}_8}$-linear Hadamard codes of type (n;t1, t2, t3) and show how to find s-PD-sets of minimum size s+1, for all s up to an upper bound, to perform a partial permutation decoding for these codes.

Abstract $${\mathbb {Z}}_{p^s}$$ Z p s -additive codes of length n are subgroups of $${\mathbb {Z}}_{p^s}^n$$ Z p s n , and can be seen as a generalization of linear codes over $${\mathbb {Z}}_2$$ Z 2 , $${\mathbb {Z}}_4$$ Z 4 , or $${\mathbb {Z}}_{2^s}$$ Z 2 s in general. A $${\mathbb {Z}}_{p^s}$$ Z p s -linear generalized Hadamard (GH) code is a GH code over $${\mathbb {Z}}_p$$ Z p which is the image of a $${\mathbb {Z}}_{p^s}$$ Z p s -additive code by a generalized Gray map. In this paper, we generalize some known results for $${\mathbb {Z}}_{p^s}$$ Z p s -linear GH codes with $$p=2$$ p = 2 to any odd prime p. First, we show some results related to the generalized Carlet’s Gray map. Then, by using an iterative construction of $${\mathbb {Z}}_{p^s}$$ Z p s -additive GH codes of type $$(n;t_1,\ldots , t_s)$$ ( n ; t 1 , … , t s ) , we show for which types the corresponding $${\mathbb {Z}}_{p^s}$$ Z p s -linear GH codes of length $$p^t$$ p t are nonlinear over $${\mathbb {Z}}_p$$ Z p . For these codes, we compute the kernel and its dimension, which allow us to give a partial classification. The obtained results for $$p\ge 3$$ p ≥ 3 are different from the case with $$p=2$$ p = 2 . Finally, the exact number of non-equivalent such codes is given for an infinite number of values of s, t, and any $$p\ge 2$$ p ≥ 2 ; by using also the rank as an invariant in some specific cases.

$\mathbb {Z}_{2^{s}}$-additive codes are subgroups of$\mathbb {Z}^{n}_{2^{s}}$, and can be seen as a generalization of linear codes over$\mathbb {Z}_{2}$and$\mathbb {Z}_{4}$. A$\mathbb {Z}_{2^{s}}$-linear code is a binary code (not necessarily linear) which is the Gray map image of a$\mathbb {Z}_{2^{s}}$-additive code. We consider$\mathbb {Z}_{2^{s}}$-additive simplex codes of type$\alpha $and$\beta $, which are a generalization over$\mathbb {Z}_{2^{s}}$of the binary simplex codes. These codes are related to the$\mathbb {Z}_{2^{s}}$-additive Hadamard codes. In this paper, we use this relationship to find a linear subcode of the corresponding$\mathbb {Z}_{2^{s}}$-linear codes, called kernel, and a representation of these codes as cosets of this kernel. In particular, this also gives the linearity of these codes. Similarly,$\mathbb {Z}_{2^{s}}$-additive MacDonald codes are defined for$s>2$, and equivalent results are obtained.

Linear codes over$\mathbb {Z}_{p^{s}}$of length$n$are subgroups of$\mathbb {Z}_{p^{s}}^{n}$. These codes are also called$\mathbb {Z}_{p^{s}}$-additive codes and can be seen as a generalization of linear codes over$\mathbb {Z}_{2}$and$\mathbb {Z}_{4}$. A$\mathbb {Z}_{p^{s}}$-linear code is a code over$\mathbb {Z}_{p}$, not necessarily linear, which is the generalized Gray map image of a$\mathbb {Z}_{p^{s}}$-additive code. In 2015, a systematic encoding was found for$\mathbb {Z}_{4}$-linear codes. Moreover, an alternative permutation decoding method, which is suitable for any binary code (not necessarily linear) with a systematic encoding, was established. In this paper, we generalize these results by presenting a systematic encoding for any$\mathbb {Z}_{p^{s}}$-linear code with$s\geq 2$and$p$prime. We also describe a permutation decoding method for any systematic code over$\mathbb {Z}_{p}$, not necessarily linear, and show some examples of how to use this systematic encoding in this decoding method.

Z2$s$-additive codes are subgroups of Zn2s, and can be seen as a generalization of linear codes over Z2and Z4. A Z2$s$-linear code is a binary code (not necessarily linear) which is the Gray map image of a Z2$s$-additive code. We consider Z2s- additive simplex codes of type a and β, which are a generalization over$Z$2$s$of the binary simplex codes. These codes are related to the Z2$s$-additive Hadamard codes. In this paper, we use this relationship to find a linear subcode of the corresponding$Z$2s-linear codes, called kernel, and a representation of these codes as cosets of this kernel. In particular, this also gives the linearity of these codes. Similarly, Z2$s$-additive MacDonald codes are defined for$s$> 2, and equivalent results are obtained.

A subset of a vector space$\mathbb {F}_{q}^{n}$is additive if it is a linear space over the field$\mathbb {F}_{p}$, where$q=p^{e}$,$p$prime, and$e>1$. Bounds on the rank and dimension of the kernel of additive generalised Hadamard (additive GH) codes are established. For specific ranks and dimensions of the kernel within these bounds, additive GH codes are constructed. Moreover, for the case$e=2$, it is shown that the given bounds are tight and it is possible to construct an additive GH code for all allowable ranks and dimensions of the kernel between these bounds. Finally, we also prove that these codes are self-orthogonal with respect to the trace Hermitian inner product, and generate pure quantum codes.

A subset of a vector space Fqnis K-additive if it is a linear space over the subfield K ⊆ Fq. Let q = pe, p prime, and e > 1. Bounds on the rank and dimension of the kernel of generalised Hadamard (GH) codes which are Fp-additive are established. For specific ranks and dimensions of the kernel within these bounds, Fp-additive GH codes are constructed. Moreover, for the case e = 2, it is shown that the given bounds are tight and it is possible to construct an Fp-additive GH code for all allowable ranks and dimensions of the kernel between these bounds. Finally, we also prove that these codes are self-orthogonal with respect to the trace Hermitian inner product, and generate pure quantum codes.

The Z2(s)-additive codes are subgroups of Zn2(s), and can be seen as a generalization of linear codes over Z2and Z4. A Z2(s)-linear Hadamard code is a binary Hadamard code which is the Gray map image of a Z2(s)-additive code. It is known that either the rank or the dimension of the kernel can be used to give a complete classification for the Z4-linear Hadamard codes. However, when s > 2, the dimension of the kernel of Z2(s)-linear Hadamard codes of length 2tonly provides a complete classification for some values of t and s. In this paper, the rank of these codes is computed for s = 3. Moreover, it is proved that this invariant, along with the dimension of the kernel, provides a complete classification, once t ≥ 3 is fixed. In this case, the number of nonequivalent such codes is also established.

A general criterion to obtain s-PD-sets of minimum size s + 1 for partial permutation decoding, which enable correction up to s errors, for systematic codes over a finite field Fq and Z4-linear codes is provided. We show how this technique can be easily applied to linear cyclic codes over Fq, Z4-linear codes which are the Gray map image of a quaternary linear cyclic code, and some related codes such as quasi-cyclic codes. Furthermore, specific results for some linear and nonlinear binary codes, including simplex, Kerdock, Delsarte-Goethals, and extended dualized Kerdock codes are given. Finally, applying this technique, new s-PD-sets of size s + 1 for Z4-linear Hadamard codes of type 2γ4δ, for all δ ≥ 4 and 1δ- 3; and for Z4-linear simplex codes of type 4m, for all m ≥ 2 and 1m+1- 3, are also provided.

Permutation decoding is a technique that strongly depends on the existence of a special subset, called PD-set, of the permutation automorphism group of a code. In this paper, a general criterion to obtain s-PD-sets of size s + 1, which enable correction up to s errors, for Z4-linear codes is provided. Furthermore, some explicit constructions of s-PD-sets of size s+1 for important families of (nonlinear) Z4-linear codes such as Hadamard and Kerdock codes are given.

The ranks and kernels of generalized Hadamard matrices are studied. It is proved that any generalized Hadamard matrix H(q, λ) over Fq, q > 3, or q = 3 and gcd(3, λ) ≠ 1, generates a self-orthogonal code. This result puts a natural upper bound on the rank of the generalized Hadamard matrices. Lower and upper bounds are given for the dimension of the kernel of the corresponding generalized Hadamard codes. For specific ranks and dimensions of the kernel within these bounds, generalized Hadamard codes are constructed.

A Z2Z4-linear Hadamard code of length α + 2β = 2tis a binary Hadamard code, which is the Gray map image of a Z2Z4-additive code with α binary coordinates and β quaternary coordinates. It is known that there are exactly ⌊t-1/2⌋ and ⌊t/2⌋ nonequivalent Z2Z4-linear Hadamard codes of length 2t, with α = 0 and α ≠ 0, respectively, for all t ≥ 3. In this paper, it is shown that each Z2Z4-linear Hadamard code with α = 0 is equivalent to a Z2Z4-linear Hadamard code with α ≠ 0, so there are only ⌊t/2⌋ nonequivalent Z2Z4-linear Hadamard codes of length 2t. Moreover, the order of the monomial automorphism group for the Z2Z4-additive Hadamard codes and the permutation automorphism group of the corresponding Z2Z4-linear Hadamard codes are given.

In a realistic distributed storage environment, like the ones used in companies dedicated to the task of storing information over a network, storage nodes are usually placed in racks, a metallic support designed to accommodate electronic equipment. It is known that the communication (bandwidth) cost between nodes which are in the same rack is much lower than between nodes which are in different racks. In this paper, a new mathematical model for a distributed storage environment where the storage nodes are placed in two racks is presented and analyzed.

The classification, up to isomorphism, of all self-embedding monomial power permutations of Hamming Steiner triple systems of order n = 2m- 1 for small m (m ≤ 22), is given. For m ∈{5, 7,11,13,17,19}, all given self-embeddings in closed surfaces are new. Moreover, they are cyclic for all m. For any non prime m, the nonexistence of such self-embeddings in a closed surface is proven. The rotation line spectrum for self-embeddings of Hamming Steiner triple systems in pseudosurfaces with pinch points as an invariant to distinguish APN permutations or, in general, to classify permutations, is proposed. This classification for APN monomial power permutations coincides with the CCZ-equivalence, at least up to m ≤ 17.

Nowadays, it is possible to optimize the storage size of large files in distributed environments, maintaining the same availability level in the system. Although replication (backups) is the most used option, it is possible to make use of erasure coding in order to significantly compress the storage size [1]. However, using erasure coding, more information needs to be transmitted than when using replication, in order to replace a node which has failed. This is called the code repair problem. The amount of transmitted information can be an important issue when the file size is very large. If network coding is used in conjunction with erasure codes, the transmitted information can be reduced by compressing the sent data in the regeneration phase. This solution minimizes the code repair problem and consists of the use of regenerating codes introduced by Dimakis et al. [2]. Nevertheless, using network coding, computational resources for any node in the system are required, since linear operations must be carried out and systems of equations must be solved in the nodes. This requirement is too strong for many simple storage devices [3]. In this paper a new family of Minimum Storage Regenerating codes is proposed. These codes not only compress the storage size using erasure coding and achieve optimality for the amount of transmitted information when d = k + 1, but also demand few computational requirements. Therefore, they could be used for simple hard discs without computational resources.

Given a 1-perfect codeC, the group of symmetries ofC,Sym(C)={π ∈Sn| π(C)=C} , is a subgroup of the group of automorphisms ofC. In this paper, we focus on symmetries of order two, i.e., involutions. LetInvF(C) ⊆Sym(C) be the set of involutions that stabilizeFpointwise. For linear 1-perfect codes, the possibilities for the number of fixed points |F| are given, establishing lower and upper bounds. For anym≥ 2 and any valuekbetween these bounds, [m/2] ≤k≤m-1, linear 1-perfect codes of lengthn=2m-1 which have an involution that fixes |F| = 2k-1 coordinates are constructed. Moreover, for anym≥ 4, 1 ≤r≤m-1, and [m/2] ≤k≤m-1, nonlinear 1-perfect codes of lengthn=2m-1 having rankn-m+rand an involution that fixes 2k-1 coordinates are also constructed, except one case, whenm≥ 6 is even,r=m-1 andk= [m/2].

Recently, new families of quaternary linear Reed–Muller codes have been introduced. They satisfy that, after the Gray map, the corresponding${\BBZ}_{4}$-linear codes have the same parameters and properties as the codes of the binary linear Reed–Muller family. A structural invariant, the dimension of the kernel, for binary codes is used to classify completely these${\BBZ}_{4}$-linear codes. The dimension of the kernel for these${\BBZ}_{4}$-linear codes is established generalizing the known results about the dimension of the kernel for${\BBZ}_{4}$-linear Hadamard and${\BBZ}_{4}$-linear extended 1-perfect codes.

The intersection structure forZ2Z4-additive Hadamard codes is investigated. For any two of these codesC1andC2, the Abelian group structure of the intersectionC1capC2is characterized. The parameters of this Abelian group structure corresponding to the intersection codes are computed, establishing lower and upper bounds for them. Constructions are given of codes whose intersection has any parameters between these bounds.

The intersection problem for Z2Z4-additive (extended and nonextended) perfect codes, i.e., which are the possibilities for the number of codewords in the intersection of two Z2Z4-additive codes C1and C2of the same length, is investigated. Lower and upper bounds for the intersection number are computed and, for any value between these bounds, codes which have this given intersection value are constructed. For all these Z2Z4-additive codes C1and C2, the abelian group structure of the intersection codes C1cap C2is characterized. The parameters of this Abelian group structure corresponding to the intersection codes are computed and lower and upper bounds for these parameters are established. Finally, for all possible parameters between these bounds, constructions of codes with these parameters for their intersections are given.

For two binary codes$C_1,C_2$, define$i(C_1,C_2)=\vert C_1 \cap C_2\vert$to be their intersection number. This correspondence establishes that there exist Hadamard codes of length$2^t$, for all$t\geq 3$, with intersection number$i$if and only if$i \in \{0,2,4,\ldots,2^{t+1}-12,2^{t+1}-8, 2^{t+1}\}$. Also it is proved that for all$t\geq 4$, if there exists a Hadamard matrix of order$4s$, then there exist Hadamard codes of length$2^{t+2}s$with intersection number$i$if and only if$i \in \{0,2,4,\ldots,2^{t+3}s-12,2^{t+3}s-8,2^{t+3}s \}$.

All the possible nonisomorphic additive (/spl Zopf//sub 4/-linear and non-/spl Zopf//sub 4/-linear) Hadamard codes are characterized and the rank and the dimension of the kernel are computed for each one.

In this paper, the rank and the dimension of the kernel for (binary) Hadamard codes of length a power of two are studied. In general, it is well-known that the rank of a Hadamard code of length n=2/sup t/ is a value in {t+1,...,n/2}. In the present paper, the range of possible values for the dimension of the kernel is computed and a construction of Hadamard codes of length n=2/sup t/ for each one of these values is given. Lower and upper bounds for the rank and dimension of the kernel of a Hadamard code of length n=2/sup t/ are also established. Finally, we construct Hadamard codes for all possible ranks and dimension of kernels between these bounds.