On [Formula Omitted]-Linear Hadamard Codes: Rank and Classification

The [Formula Omitted]-additive codes are subgroups of [Formula Omitted], and can be seen as a generalization of linear codes over [Formula Omitted] and [Formula Omitted]. A [Formula Omitted]-linear Hadamard code is a binary Hadamard code which is the Gray map image of a [Formula Omitted]-additive co...

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Bibliographic Details
Published in:IEEE transactions on information theory 2020-01, Vol.66 (2), p.970
Main Authors: Fernandez-Cordoba, Cristina, Vela, Carlos, Villanueva, Merce
Format: Article
Language:English
Subjects:
Binary codes
Classification
Codes
Error correcting codes
Kernels
Subgroups
Online Access:Get full text
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Summary:The [Formula Omitted]-additive codes are subgroups of [Formula Omitted], and can be seen as a generalization of linear codes over [Formula Omitted] and [Formula Omitted]. A [Formula Omitted]-linear Hadamard code is a binary Hadamard code which is the Gray map image of a [Formula Omitted]-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 [Formula Omitted]-linear Hadamard codes. However, when [Formula Omitted], the dimension of the kernel of [Formula Omitted]-linear Hadamard codes of length [Formula Omitted] only provides a complete classification for some values of [Formula Omitted] and [Formula Omitted]. In this paper, the rank of these codes is computed for [Formula Omitted]. Moreover, it is proved that this invariant, along with the dimension of the kernel, provides a complete classification, once [Formula Omitted] is fixed. In this case, the number of nonequivalent such codes is also established.
ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2019.2952599