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Mathematics 2017, 5(4), 82; https://doi.org/10.3390/math5040082

Multiplicative Structure and Hecke Rings of Generator Matrices for Codes over Quotient Rings of Euclidean Domains

Toyota Technological Institute, 2–12–1 Hisakata, Tempaku, Nagoya, Aichi 468–8511, Japan
Received: 30 June 2017 / Revised: 20 November 2017 / Accepted: 8 December 2017 / Published: 15 December 2017
(This article belongs to the Special Issue Geometry of Numbers)
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Abstract

In this study, we consider codes over Euclidean domains modulo their ideals. In the first half of the study, we deal with arbitrary Euclidean domains. We show that the product of generator matrices of codes over the rings mod a and mod b produces generator matrices of all codes over the ring mod a b , i.e., this correspondence is onto. Moreover, we show that if a and b are coprime, then this correspondence is one-to-one, i.e., there exist unique codes over the rings mod a and mod b that produce any given code over the ring mod a b through the product of their generator matrices. In the second half of the study, we focus on the typical Euclidean domains such as the rational integer ring, one-variable polynomial rings, rings of Gaussian and Eisenstein integers, p-adic integer rings and rings of one-variable formal power series. We define the reduced generator matrices of codes over Euclidean domains modulo their ideals and show their uniqueness. Finally, we apply our theory of reduced generator matrices to the Hecke rings of matrices over these Euclidean domains. View Full-Text
Keywords: error-correcting codes; quasi-cyclic codes; Euclidean division; Hermite normal form; Hecke algebras error-correcting codes; quasi-cyclic codes; Euclidean division; Hermite normal form; Hecke algebras
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).
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Matsui, H. Multiplicative Structure and Hecke Rings of Generator Matrices for Codes over Quotient Rings of Euclidean Domains. Mathematics 2017, 5, 82.

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