A Hadamard Product of Linear Codes: Algebraic Properties and Algorithms for Calculating It
Moscow University Computational Mathematics and Cybernetics
Pub Date : 2024-02-04
DOI:10.3103/s0278641923040179
引用次数: 0
Abstract
A study is performed of the algebraic properties of the Hadamard product (Schur product, component-wise product) of linear error-correcting codes. The complexity of constructing a product basis using known multiplier bases is discussed. The concept is introduced of quotient, quasi-quotient, and maximal inclusion quasi-quotient obtained from the Hadamard division of one linear code by another. An explicit form of the maximum Hadamard division quasi-quotient is established. A criterion is proved for the existence of a given code of an inverse code in a semiring formed by linear codes of length \(n\) with the operations of sum and product of Hadamard codes. The explicit form of codes that have a reverse code in this semiring is described.
线性编码的哈达玛积：代数特性和计算算法
摘要 研究了线性纠错码的哈达玛乘积（舒尔乘积，分量-明智乘积）的代数性质。讨论了使用已知乘法基构建积基的复杂性。引入了从一种线性编码对另一种线性编码的哈达玛除法中获得的商、准商和最大包含准商的概念。建立了最大哈达玛除法准商的明确形式。证明了在一个由长度为 \(n\) 的线性编码与哈达玛编码的和与积运算形成的配系中存在给定编码的逆编码的标准。描述了在这个配系中具有反码的编码的显式。
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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