Characteristics of invariant weights related to code equivalence over rings. (English) Zbl 1259.94068

Summary: 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, generalizing the approach taken in M. Greferath and T. Honold [“Monomial extensions of isometries of linear codes. II: Invariant weight functions on \(Z_m\)”, in: Proceedings of the tenth international workshop in algebraic and combinatorial coding theory, ACCT-10, Zvenigorod, Russia, 106–111 (2006)].


94B05 Linear codes (general theory)
11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
05E99 Algebraic combinatorics
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