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$\mathbb{Z}_{2}\mathbb{Z}_{4}\mathbb{Z}_{8}$-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$2^{t}$.

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.