On the Kernel of ℤ_2^s-Linear Hadamard Codes

2018/01/16 by Fernández-Córdoba, Cristina, Vela, Carlos, Villanueva, Mercè
#94B25 and 94B60 #FOS: Computer and information sciences #Information Theory (cs.IT)

paper · doi:10.48550/arxiv.1801.05189

Abstract

The ℤ_2^s-additive codes are subgroups of ℤⁿ_2^s, and can be seen as a generalization of linear codes over ℤ₂ and ℤ₄. A ℤ_2^s-linear Hadamard code is a binary Hadamard code which is the Gray map image of a ℤ_2^s-additive code. It is known that the dimension of the kernel can be used to give a complete classification of the ℤ₄-linear Hadamard codes. In this paper, the kernel of ℤ_2^s-linear Hadamard codes and its dimension are established for s > 2. Moreover, we prove that this invariant only provides a complete classification for some values of t and s. The exact amount of nonequivalent such codes are given up to t=11 for any s≥ 2, by using also the rank and, in some cases, further computations.

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