Characteristics of invariant weights related to code equivalence over rings

Authors:
Marcus Greferath;Cathy Mc Fadden;Jens Zumbrägel
Affiliations:
School of Mathematical Sciences, University College Dublin, Dublin, Republic of Ireland and Claude Shannon Institute for Discrete Mathematics, Coding, Cryptography, and Information Security, Dubli ...;School of Mathematical Sciences, University College Dublin, Dublin, Republic of Ireland and Claude Shannon Institute for Discrete Mathematics, Coding, Cryptography, and Information Security, Dubli ...;School of Mathematical Sciences, University College Dublin, Dublin, Republic of Ireland and Claude Shannon Institute for Discrete Mathematics, Coding, Cryptography, and Information Security, Dubli ...
Venue:
Designs, Codes and Cryptography
Year:
2013

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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).