On generator and parity-check polynomial matrices of generalized quasi-cyclic codes

Matsui, Hajime

doi:10.1016/j.ffa.2015.02.003

Cited by 16 publications

(25 citation statements)

References 26 publications

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“…By applying the division algorithm to G in Proposition 2, we obtain the systematic encoding of C, which is similar to that in the case of GQC codes [5].…”

Section: Cardinality Formulaementioning

confidence: 88%

“…In [5], the author identified C with F −1 (C) and expressed them using the same notation C. In this study, we distinguish them and use the notation C only for an R-module in M.…”

Section: R-modules In Mmentioning

confidence: 99%

“…In this study, we need the following division algorithm, which is described in [5] in the special case where…”

Section: Euclidean Division By a Class Of Matricesmentioning

confidence: 99%

“…The structural properties of quasi-cyclic (QC) [1,2] and generalized quasi-cyclic (GQC) codes [3][4][5] have been reported. On the other hand, cyclic codes can be extended to pseudo-cyclic (PC) and generalized pseudo-cyclic (GPC) codes [6].…”

Section: Introductionmentioning

confidence: 99%

“…On the other hand, cyclic codes can be extended to pseudo-cyclic (PC) and generalized pseudo-cyclic (GPC) codes [6]. Similar constructions for the rational integer ring Z are known as integer codes and generalized integer codes [7].…”

Section: Introductionmentioning

confidence: 99%

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

Matsui

2017

Mathematics

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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 ab, 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 ab 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.

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“…By applying the division algorithm to G in Proposition 2, we obtain the systematic encoding of C, which is similar to that in the case of GQC codes [5].…”

Section: Cardinality Formulaementioning

confidence: 88%

“…In [5], the author identified C with F −1 (C) and expressed them using the same notation C. In this study, we distinguish them and use the notation C only for an R-module in M.…”

Section: R-modules In Mmentioning

confidence: 99%

“…In this study, we need the following division algorithm, which is described in [5] in the special case where…”

Section: Euclidean Division By a Class Of Matricesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Matsui

2017

Mathematics

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Two-sided Galois duals of multi-twisted codes

ElDin

2023

J. Appl. Math. Comput.

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Galois duals of Multi-twisted (MT) codes are considered in this study. We describe a MT code $$\mathcal {C}$$ C as a module over a principal ideal domain. Hence, $$\mathcal {C}$$ C has a generator polynomial matrix (GPM) $$\textbf{G}$$ G that satisfies an identical equation. We prove a GPM formula for the Euclidean dual $$\mathcal {C}^\perp $$ C ⊥ using the identical equation of the Hermite normal form of $$\textbf{G}$$ G . Next, we aim to replace the Euclidean dual with the Galois dual. The Galois inner product is an asymmetric form, so we distinguish between the right and left Galois duals. We show that the right and left Galois duals of a MT code are MT as well but with possibly different shift constants. Some interconnected identities for the right and left Galois duals of a linear code are established and we also introduce the two-sided Galois dual. We use a condition that makes the two-sided Galois dual of a MT code MT, then we describe its GPM. Two special cases are also studied, one when the right and left Galois duals trivially intersect and the other when they coincide. A necessary and sufficient condition is established for the equality of the right and left Galois duals of any linear code.

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