On \mathbb} -Linear Hadamard Codes: Rank and Classification

• Published in: IEEE transactions on information theory 2020-02, Vol.66 (2), p.970-982
• Main Authors: Fernandez-Cordoba, Cristina, Vela, Carlos, Villanueva, Merce
• Format: Article
• Language: English
• Subjects: Additives; Binary codes; classification; Error correction; Error correction codes; Gray map; Hadamard code; II-VI semiconductor materials; Kernel; Linear codes; Rank; Z₂<italic xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">s -additive code
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2019.2952599

The \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 Hadamard code is a binary Hadamard code which is the Gray map image of a \mathbb {Z}_{2^{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 \mathbb {Z}_{4} -linear Hadamard codes. However, when s > 2 , the dimension of the kernel of \mathbb {Z}_{2^{s}} -linear Hadamard codes of length 2^{t} only 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\geq 3 is fixed. In this case, the number of nonequivalent such codes is also established.