Families of hadamard Z<inf>2Z4</inf>Z<inf>8</inf>-codes

Angel Del Rio; Josep Rifa

doi:10.1109/TIT.2013.2258373

Families of hadamard Z<inf>2Z4</inf>Z<inf>8</inf>-codes

Angel Del Rio, Josep Rifa

Research output: Contribution to journal › Article › Research › peer-review

5 Citations (Scopus)

Abstract

A Z2Z4Q8-code is the binary image, after a Gray map, of a subgroup of BBZ2k-1×\BBZ4k 2× Q8 k3, where Q8 is the quaternion group on eight elements. Such BBZ2\BBZ4Q8-codes are translation invariant propelinear codes as are the well known BBZ 4-linear or BBZ2BBZ4-linear codes. In this paper, we show that there exist 'pure' BBZ-2 BBZ4Q 8-codes, that is, codes that do not admit any abelian translation invariant propelinear structure. We study the dimension of the kernel and rank of the BBZ 2\BBZ4Q8-codes, and we give upper and lower bounds for these parameters. We give tools to construct a new class of Hadamard codes formed by several families of BBZ2\BBZ 4Q8-codes; we classify such codes from an algebraic point of view and we improve the upper and lower bounds for the rank and the dimension of the kernel when the codes are Hadamard. © 2013 IEEE.

Original language	English
Article number	6508950
Pages (from-to)	5140-5151
Journal	IEEE Transactions on Information Theory
Volume	59
Issue number	8
DOIs	https://doi.org/10.1109/TIT.2013.2258373
Publication status	Published - 29 Jul 2013

Keywords

1-perfect codes
BBZ BBZ Q -codes 2 4 8
Hadamard codes
propelinear codes
translation invariant codes

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10.1109/TIT.2013.2258373

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@article{74c57ec099fe4a59a3e6f95b4ec1e4cf,

title = "Families of hadamard Z2Z4Z8-codes",

abstract = "A Z2Z4Q8-code is the binary image, after a Gray map, of a subgroup of BBZ2k-1×\BBZ4k 2× Q8 k3, where Q8 is the quaternion group on eight elements. Such BBZ2\BBZ4Q8-codes are translation invariant propelinear codes as are the well known BBZ 4-linear or BBZ2BBZ4-linear codes. In this paper, we show that there exist 'pure' BBZ-2 BBZ4Q 8-codes, that is, codes that do not admit any abelian translation invariant propelinear structure. We study the dimension of the kernel and rank of the BBZ 2\BBZ4Q8-codes, and we give upper and lower bounds for these parameters. We give tools to construct a new class of Hadamard codes formed by several families of BBZ2\BBZ 4Q8-codes; we classify such codes from an algebraic point of view and we improve the upper and lower bounds for the rank and the dimension of the kernel when the codes are Hadamard. {\textcopyright} 2013 IEEE.",

keywords = "1-perfect codes, BBZ BBZ Q -codes 2 4 8, Hadamard codes, propelinear codes, translation invariant codes",

author = "{Del Rio}, Angel and Josep Rifa",

year = "2013",

month = jul,

day = "29",

doi = "10.1109/TIT.2013.2258373",

language = "English",

volume = "59",

pages = "5140--5151",

number = "8",

}

TY - JOUR

T1 - Families of hadamard Z2Z4Z8-codes

AU - Del Rio, Angel

AU - Rifa, Josep

PY - 2013/7/29

Y1 - 2013/7/29

N2 - A Z2Z4Q8-code is the binary image, after a Gray map, of a subgroup of BBZ2k-1×\BBZ4k 2× Q8 k3, where Q8 is the quaternion group on eight elements. Such BBZ2\BBZ4Q8-codes are translation invariant propelinear codes as are the well known BBZ 4-linear or BBZ2BBZ4-linear codes. In this paper, we show that there exist 'pure' BBZ-2 BBZ4Q 8-codes, that is, codes that do not admit any abelian translation invariant propelinear structure. We study the dimension of the kernel and rank of the BBZ 2\BBZ4Q8-codes, and we give upper and lower bounds for these parameters. We give tools to construct a new class of Hadamard codes formed by several families of BBZ2\BBZ 4Q8-codes; we classify such codes from an algebraic point of view and we improve the upper and lower bounds for the rank and the dimension of the kernel when the codes are Hadamard. © 2013 IEEE.

AB - A Z2Z4Q8-code is the binary image, after a Gray map, of a subgroup of BBZ2k-1×\BBZ4k 2× Q8 k3, where Q8 is the quaternion group on eight elements. Such BBZ2\BBZ4Q8-codes are translation invariant propelinear codes as are the well known BBZ 4-linear or BBZ2BBZ4-linear codes. In this paper, we show that there exist 'pure' BBZ-2 BBZ4Q 8-codes, that is, codes that do not admit any abelian translation invariant propelinear structure. We study the dimension of the kernel and rank of the BBZ 2\BBZ4Q8-codes, and we give upper and lower bounds for these parameters. We give tools to construct a new class of Hadamard codes formed by several families of BBZ2\BBZ 4Q8-codes; we classify such codes from an algebraic point of view and we improve the upper and lower bounds for the rank and the dimension of the kernel when the codes are Hadamard. © 2013 IEEE.

KW - 1-perfect codes

KW - BBZ BBZ Q -codes 2 4 8

KW - Hadamard codes

KW - propelinear codes

KW - translation invariant codes

U2 - 10.1109/TIT.2013.2258373

DO - 10.1109/TIT.2013.2258373

M3 - Article

SN - 0018-9448

VL - 59

SP - 5140

EP - 5151

JO - IEEE Transactions on Information Theory

JF - IEEE Transactions on Information Theory

IS - 8

M1 - 6508950

ER -

Families of hadamard Z<inf>2Z4</inf>Z<inf>8</inf>-codes

Abstract

Keywords

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