Linear complementary pairs of codes over a finite non-commutative Frobenius ring

IF 2.4 3区数学 Q1 MATHEMATICS

Journal of Applied Mathematics and Computing Pub Date : 2024-06-24 DOI:10.1007/s12190-024-02161-w

Sanjit Bhowmick, Xiusheng Liu

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Abstract

In this paper, we study linear complementary pairs (LCP) of codes over finite non-commutative local rings. We further provide a necessary and sufficient condition for a pair of codes (C, D) to be LCP of codes over finite non-commutative Frobenius rings. The minimum distances d(C) and \(d(D^\perp )\) are defined as the security parameter for an LCP of codes (C, D). It was recently demonstrated that if C and D are both 2-sided LCP of group codes over a finite commutative Frobenius rings, \(D^\perp \) and C are permutation equivalent in Liu and Liu (Des Codes Cryptogr 91:695–708, 2023). As a result, the security parameter for a 2-sided group LCP (C, D) of codes is simply d(C). Towards this, we deliver an elementary proof of the fact that for a linear complementary pair of codes (C, D), where C and D are linear codes over finite non-commutative Frobenius rings, under certain conditions, the dual code \(D^\perp \) is equivalent to C.

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有限非交换弗罗本尼斯环上的线性互补对码

本文研究了有限非交换局部环上的编码线性互补对（LCP）。我们进一步提供了一对码（C, D）成为有限非交换弗罗贝尼斯环上的线性互补码的必要条件和充分条件。最小距离 d(C) 和 \(d(D^\perp )\) 被定义为代码 (C, D) 的 LCP 的安全参数。最近，Liu 和 Liu (Des Codes Cryptogr 91:695-708, 2023)证明，如果 C 和 D 都是有限交换弗罗贝尼斯环上群码的双面 LCP，那么 \(D^\perp \) 和 C 是等价的。因此，代码的双面组 LCP (C, D) 的安全参数就是 d(C)。为此，我们提供了一个基本证明：对于一对线性互补码（C, D），其中 C 和 D 是有限非交换弗罗贝尼斯环上的线性码，在某些条件下，对偶码 \(D^\perp \) 等同于 C。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Applied Mathematics and Computing Mathematics-Computational Mathematics

CiteScore

4.20

自引率

4.50%

发文量

131

期刊介绍： JAMC is a broad based journal covering all branches of computational or applied mathematics with special encouragement to researchers in theoretical computer science and mathematical computing. Major areas, such as numerical analysis, discrete optimization, linear and nonlinear programming, theory of computation, control theory, theory of algorithms, computational logic, applied combinatorics, coding theory, cryptograhics, fuzzy theory with applications, differential equations with applications are all included. A large variety of scientific problems also necessarily involve Algebra, Analysis, Geometry, Probability and Statistics and so on. The journal welcomes research papers in all branches of mathematics which have some bearing on the application to scientific problems, including papers in the areas of Actuarial Science, Mathematical Biology, Mathematical Economics and Finance.