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.

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.

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.

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.

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 ℤ2ℤ4-additive code C ⊆ ℤ2α× ℤ4βis called cyclic if the set of coordinates can be partitioned into two subsets, the set of ℤ2coordinates and the set of ℤ4coordinates, such that any cyclic shift of the coordinates of both subsets leaves the code invariant. Let Φ(C) be the binary Gray map image of C. We study the rank and the dimension of the kernel of a ℤ2ℤ4-additive cyclic code C, that is, the dimensions of the binary linear codes (Φ(C)) and ker (Φ(C)). We give upper and lower bounds for these parameters. It is known that the codes (Φ(C)) and ker (Φ(C)) are binary images of ℤ2ℤ4-additive codes that we denote by R(C) and K(C), respectively. Moreover, we show that R(C) and K(C) are also cyclic and determine the generator polynomials of these codes in terms of the generator polynomials of the code C.

A 762764-additive code C ⊆ ℤ2α× ℤ4βis called cyclic if the set of coordinates can be partitioned into two subsets, the set of ℤ2and the set of ℤ4coordinates, such that any cyclic shift of the coordinates of both subsets leaves the code invariant. We study the binary images of ℤ2ℤ4-additive cyclic codes. We determine all ℤ2ℤ4-additive cyclic codes with odd β whose Gray images are linear binary codes. In this case, it is shown that such binary codes are permutation equivalent (by the Nechaev permutation) to ℤ2-double cyclic codes. Finally, the generator polynomials of these binary codes are given.

A ℤ2ℤ4-additive code C ⊆ ℤ2αx ℤ4βis called cyclic if the set of coordinates can be partitioned into two subsets, the set of ℤ2and the set of ℤ4coordinates, such that any cyclic shift of the coordinates of both subsets leaves the code invariant. These codes can be identified as submodules of the ℤ4[x]-module ℤ2[x]/(xα- 1) x ℤ4[x]/(xβ- 1). The parameters of a ℤ2ℤ4-additive cyclic code are stated in terms of the degrees of the generator polynomials of the code. The generator polynomials of the dual code of a ℤ2ℤ4-additive cyclic code are determined in terms of the generator polynomials of the code C.

Following [5], given a Z2Z4-additive self-dual code, one can easily extend this code and generate an extended Z2Z4-additive self-dual code with greater length. In this communication we study these constructions and check if properties like separability and code Type are retained or not.

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].

We describe recent results for codes over Z2×Z4giving their connection to binary codes via a natural Gray map. We study Z2Z4self-dual codes and we state the major results concerning these codes. We state several open questions and discuss possible avenues of research.

Quaternary ZRM(r,m) codes were defined so that their binary images, via Gray map, are Reed-Muller codes for some specific values of . In the literature, two different definitions of such codes can be found. They will be denoted ZRM(r,m) and ZRM-(r,m) codes. In this correspondence, we show that both definitions are equivalent exactly for those values of r such that their binary images are Reed-Muller codes. Moreover, we prove that, for all r, these binary images are linear codes in the case of ZRM(r,m), but they are not if we use the definition of ZRM-(r,m). In this last case, we compute the rank and the dimension of the kernel of these codes.