On the linearity and classification of $${\mathbb {Z}}_{p^s}$$-linear generalized hadamard codes

Abstract

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.

Construction and Linearity of Z_pZ_{p^2}-Linear Generalized Hadamard Codes

“…Finally, in Section 6, we show some computational results for p = 3 and p = 5, which point out that, unlike Z 2 Z 4 -linear GH codes, when p ≥ 3 prime, the Z p 2 -linear GH codes are not included in the family of Z p Z p 2 -linear GH codes with α 1 = 0. Moreover, we also observe that they are not equivalent to any of the Z p s -linear GH codes considered in [7,15] by using the same Gray map.…”

“…Moreover, it is also known that the family of Z 2 Z 4 -linear Hadamard codes with α 1 = 0 includes the family of Z 2 Z 4 -linear Hadamard codes with α 1 = 0 [14], since each Z 2 Z 4linear Hadamard code with α 1 = 0 is equivalent to a Z 2 Z 4 -linear Hadamard code with α 1 = 0. The rank and dimension of the kernel have also been used to classify Z p s -linear GH codes of length p t , with p prime [7,16,17,15].…”

“…Moreover, in [14], it is shown that each Z 2 Z 4 -linear GH code with α 1 = 0 is equivalent to a Z 2 Z 4 -linear GH code with α 1 = 0, so indeed there are only ⌊t/2⌋ non-equivalent Z 2 Z 4 -linear GH codes of length 2 t . Later, in [7,15,16,17], an iterative construction for Z p s -linear GH codes is described, the linearity is established, and a partial classification is obtained, giving the exact amount of non-equivalent non-linear such codes for some parameters.…”

“…This paper is focused on Z p Z p 2 -linear GH codes with α 1 = 0 and p ≥ 3 prime, generalizing some results given for p = 2 in [13,18] related to the construction and linearity of such codes. For p = 3 and 2 ≤ t ≤ 8, these codes are compared with the Z p Z p 2 -linear GH codes of length p t with α 1 = 0 studied in [15]. This paper is organized as follows.…”

Equivalences among Z_{p^s}-linear Generalized Hadamard Codes

“…Finally, for s ∈ {2, 3}, they establish the full classification of the Z 2 s -linear Hadamard codes of length 2 t by giving the exact number of such codes. For s ≥ 2, and p ≥ 3 prime, the dimension of the kernel for Z p s -linear GH codes of length p t is established in [6], and it is proved that this invariant only provides a complete classification for certain values of t and s. Lower and upper bounds are also established for the number of nonequivalent Z p s -linear GH codes of length p t , when both t and s are fixed, and when just t is fixed; denoted by A t,s,p and A t,p , respectively. From [6], we can check that there are nonlinear codes having the same rank and dimension of the kernel for different values of s, once the length p t is fixed, for all 4 ≤ t ≤ 11.…”