Two approaches to the extension problem for arbitrary weights over finite module alphabets
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Two approaches to the extension problem for arbitrary weights over finite module alphabets Jay A. Wood1 Received: 20 January 2020 / Accepted: 10 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract The extension problem underlies notions of code equivalence. Two approaches to the extension problem are described. One is a matrix approach that reduces the gen‑ eral problem for weights to one for symmetrized weight compositions. The other is a monoid algebra approach that reframes the extension problem in terms of modules over the monoid algebra determined by the multiplicative monoid of a finite ring. Keywords Frobenius ring · Linear code · Extension theorem · Monoid algebra · Cyclic socle · Symmetrized weight composition Mathematics Subject Classification Primary 94B05 · Secondary 16S36, 20M25
1 Introduction The extension problem has been a fascination of mine ever since Vera Pless sug‑ gested, on 28 April 1992, that I study MacWilliams’s theorem on code equivalence [18, 19]. Many papers on the extension problem have been published since that time. The present paper attempts to synthesize earlier results and to present an approach to the extension problem using monoid algebras that may help clarify the structures that underlie the problem. The extension problem arises from the question: When should two linear codes be considered equivalent? There are two reasonable responses: when there is a monomial transformation of the ambient space that maps one code to the other; and when there is a linear isometry between the two codes. Equivalence in the first sense implies equivalence in the second sense. The converse is the extension problem: if
In memoriam: Florence E. Wood, 1922–2019. Mary Elizabeth Keyte Moore, 1932–2019. * Jay A. Wood [email protected] 1
Western Michigan University, 1903 W. Michigan Ave., Kalamazoo, MI 49008‑5248, USA
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f ∶ C1 → C2 is a linear isometry, does f extend to a linear isometry/monomial trans‑ formation of the ambient space? Left unsaid in the preceding paragraph is the exact context of the extension problem. There is an alphabet A and a weight w on A. In this paper, the alphabet A will be a finite unital left module over a finite ring R with 1. A linear code C is a left R-submodule of An , which is the ambient space. The weight w is a function w ∶ A → ℂ , the complex numbers, satisfying w(0) = 0 . Then w is extended addi‑ ∑ tively to a weight on An : w(a1 , a2 , … , an ) = ni=1 w(ai ) . (There are weights on An that are not of this form, but we do not study them here.) MacWilliams [18, 19] solved the extension problem for the Hamming weight over finite fields. Subsequent results have solved the extension problem in increas‑ ingly broader contexts. For example: for the Hamming weight [21] and the homoge‑ neous weight [12] over finite Frobenius rings; for the Hamming and homogeneous weights over Frobenius bimodules [11] and over finite modules that are pseudoinjective and have a cyclic socle [24]. For weights beyond th
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