Characteristics of invariant weights related to code equivalence over rings

Marcus Greferath; Cathy Mc Fadden; Jens Zumbrägel

doi:10.1007/s10623-012-9671-9

Characteristics of invariant weights related to code equivalence over rings

Marcus Greferath, Cathy Mc Fadden, Jens Zumbrägel

Not published at Aalto University

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

4 Citations (Scopus)

Abstract

The Equivalence Theorem states that, for a given weight on an alphabet, every isometry between linear codes extends to a monomial transformation of the entire space. This theorem has been proved for several weights and alphabets, including the original MacWilliams’ Equivalence Theorem for the Hamming weight on codes over finite fields. The question remains: What conditions must a weight satisfy so that the Extension Theorem will hold? In this paper we provide an algebraic framework for determining such conditions, generalising the approach taken in Greferath and Honold (Proceedings of the Tenth International Workshop in Algebraic and Combinatorial Coding Theory (ACCT-10), pp. 106–111. Zvenigorod, Russia, 2006).

Original language	Undefined/Unknown
Pages (from-to)	145-156
Number of pages	12
Journal	DESIGNS CODES AND CRYPTOGRAPHY
Volume	66
Issue number	1-3
DOIs	https://doi.org/10.1007/s10623-012-9671-9
Publication status	Published - 2013
MoE publication type	A1 Journal article-refereed

Keywords

MacWilliams’ equivalence theorem
extension theorem
weights
ring-linear codes

Access to Document

10.1007/s10623-012-9671-9

http://dx.doi.org/10.1007/s10623-012-9671-9

Cite this

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title = "Characteristics of invariant weights related to code equivalence over rings",

abstract = "The Equivalence Theorem states that, for a given weight on an alphabet, every isometry between linear codes extends to a monomial transformation of the entire space. This theorem has been proved for several weights and alphabets, including the original MacWilliams{\textquoteright} Equivalence Theorem for the Hamming weight on codes over finite fields. The question remains: What conditions must a weight satisfy so that the Extension Theorem will hold? In this paper we provide an algebraic framework for determining such conditions, generalising the approach taken in Greferath and Honold (Proceedings of the Tenth International Workshop in Algebraic and Combinatorial Coding Theory (ACCT-10), pp. 106–111. Zvenigorod, Russia, 2006).",

keywords = "MacWilliams{\textquoteright} equivalence theorem, extension theorem, weights, ring-linear codes",

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pages = "145--156",

journal = "DESIGNS CODES AND CRYPTOGRAPHY",

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TY - JOUR

T1 - Characteristics of invariant weights related to code equivalence over rings

AU - Greferath, Marcus

AU - Mc Fadden, Cathy

AU - Zumbrägel, Jens

PY - 2013

Y1 - 2013

N2 - The Equivalence Theorem states that, for a given weight on an alphabet, every isometry between linear codes extends to a monomial transformation of the entire space. This theorem has been proved for several weights and alphabets, including the original MacWilliams’ Equivalence Theorem for the Hamming weight on codes over finite fields. The question remains: What conditions must a weight satisfy so that the Extension Theorem will hold? In this paper we provide an algebraic framework for determining such conditions, generalising the approach taken in Greferath and Honold (Proceedings of the Tenth International Workshop in Algebraic and Combinatorial Coding Theory (ACCT-10), pp. 106–111. Zvenigorod, Russia, 2006).

AB - The Equivalence Theorem states that, for a given weight on an alphabet, every isometry between linear codes extends to a monomial transformation of the entire space. This theorem has been proved for several weights and alphabets, including the original MacWilliams’ Equivalence Theorem for the Hamming weight on codes over finite fields. The question remains: What conditions must a weight satisfy so that the Extension Theorem will hold? In this paper we provide an algebraic framework for determining such conditions, generalising the approach taken in Greferath and Honold (Proceedings of the Tenth International Workshop in Algebraic and Combinatorial Coding Theory (ACCT-10), pp. 106–111. Zvenigorod, Russia, 2006).

KW - MacWilliams’ equivalence theorem

KW - extension theorem

KW - weights

KW - ring-linear codes

U2 - 10.1007/s10623-012-9671-9

DO - 10.1007/s10623-012-9671-9

M3 - Article

VL - 66

SP - 145

EP - 156

JO - DESIGNS CODES AND CRYPTOGRAPHY

JF - DESIGNS CODES AND CRYPTOGRAPHY

SN - 0925-1022

IS - 1-3

ER -