On Z 2s -linear Hadamard codes: kernel and partial classification

Cristina Fernández-Córdoba; Carlos Vela; Mercè Villanueva

doi:10.1007/s10623-018-0546-6

On Z 2s -linear Hadamard codes: kernel and partial classification

Cristina Fernández-Córdoba, Carlos Vela, Mercè Villanueva

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7 Citations (Scopus)

Abstract

The Z2s-additive codes are subgroups of Z2sn, and can be seen as a generalization of linear codes over Z 2 and Z 4 . A Z2s-linear Hadamard code is a binary Hadamard code which is the Gray map image of a Z2s-additive code. It is known that the dimension of the kernel can be used to give a complete classification of the Z 4 -linear Hadamard codes. In this paper, the kernel of Z2s-linear Hadamard codes of length 2 t 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.

Original language	English
Pages (from-to)	417-435
Number of pages	19
Journal	Designs, Codes, and Cryptography
Volume	87
Issue number	2-3
DOIs	https://doi.org/10.1007/s10623-018-0546-6
Publication status	Published - 15 Mar 2019

Keywords

Classification
Gray map
Hadamard code
Kernel
Z2s-additive code
Z2s-linear code

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10.1007/s10623-018-0546-6

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title = "On Z 2s -linear Hadamard codes: kernel and partial classification",

abstract = "The Z2s-additive codes are subgroups of Z2sn, and can be seen as a generalization of linear codes over Z 2 and Z 4 . A Z2s-linear Hadamard code is a binary Hadamard code which is the Gray map image of a Z2s-additive code. It is known that the dimension of the kernel can be used to give a complete classification of the Z 4 -linear Hadamard codes. In this paper, the kernel of Z2s-linear Hadamard codes of length 2 t 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.",

keywords = "Classification, Gray map, Hadamard code, Kernel, Z2s-additive code, Z2s-linear code",

author = "Cristina Fern{\'a}ndez-C{\'o}rdoba and Carlos Vela and Merc{\`e} Villanueva",

note = "Funding Information: Acknowledgements This work has been partially supported by the Spanish MINECO under Grants TIN2016-77918-P (AEI/FEDER, UE) and MTM2015-69138-REDT. Publisher Copyright: {\textcopyright} 2018, Springer Science+Business Media, LLC, part of Springer Nature. Copyright: Copyright 2019 Elsevier B.V., All rights reserved.",

year = "2019",

month = mar,

day = "15",

doi = "10.1007/s10623-018-0546-6",

language = "English",

volume = "87",

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TY - JOUR

T1 - On Z 2s -linear Hadamard codes: kernel and partial classification

AU - Fernández-Córdoba, Cristina

AU - Vela, Carlos

AU - Villanueva, Mercè

N1 - Funding Information: Acknowledgements This work has been partially supported by the Spanish MINECO under Grants TIN2016-77918-P (AEI/FEDER, UE) and MTM2015-69138-REDT. Publisher Copyright: © 2018, Springer Science+Business Media, LLC, part of Springer Nature. Copyright: Copyright 2019 Elsevier B.V., All rights reserved.

PY - 2019/3/15

Y1 - 2019/3/15

N2 - The Z2s-additive codes are subgroups of Z2sn, and can be seen as a generalization of linear codes over Z 2 and Z 4 . A Z2s-linear Hadamard code is a binary Hadamard code which is the Gray map image of a Z2s-additive code. It is known that the dimension of the kernel can be used to give a complete classification of the Z 4 -linear Hadamard codes. In this paper, the kernel of Z2s-linear Hadamard codes of length 2 t 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.

AB - The Z2s-additive codes are subgroups of Z2sn, and can be seen as a generalization of linear codes over Z 2 and Z 4 . A Z2s-linear Hadamard code is a binary Hadamard code which is the Gray map image of a Z2s-additive code. It is known that the dimension of the kernel can be used to give a complete classification of the Z 4 -linear Hadamard codes. In this paper, the kernel of Z2s-linear Hadamard codes of length 2 t 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.

KW - Classification

KW - Gray map

KW - Hadamard code

KW - Kernel

KW - Z2s-additive code

KW - Z2s-linear code

UR - http://www.mendeley.com/research/z-2s-linear-hadamard-codes-kernel-partial-classification

U2 - 10.1007/s10623-018-0546-6

DO - 10.1007/s10623-018-0546-6

M3 - Article

VL - 87

SP - 417

EP - 435

JO - Designs, Codes, and Cryptography

JF - Designs, Codes, and Cryptography

SN - 0925-1022

IS - 2-3

ER -

On Z 2s -linear Hadamard codes: kernel and partial classification

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Keywords

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