基于一般仿射群正则子群的q元线性完美码

IF 0.5 4区计算机科学 Q4 COMPUTER SCIENCE, THEORY & METHODS

Problems of Information Transmission Pub Date : 2022-04-10 DOI:10.1134/s0032946022010045

I. Yu. Mogilnykh

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引用次数: 0

摘要

如果一个码的自同构群包含一个有规律地作用于码字的子群，那么这个码就是线性的。如果仿射变换的群\(GA(r,q)\)的子群有规律地作用于\(\mathbb{F}_q^r\)的向量，则称它是正则的。一般仿射群\(GA(r,q)\)的正则子群的每一个自同构都在长度为\(\frac{q^r-1}{q-1}\)的汉明码的集上引起一个置换。在此基础上，我们提出了一个长度为\(\frac{q^{r+1}-1}{q-1}\)的\(q\) - - - - -任意的线性完美码的构造。特别地，对于任意素数\(q\)，我们得到了一个几乎满秩的无穷级数\(q\) -任意线性完美码。

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On q-ary Propelinear Perfect Codes Based on Regular Subgroups of the General Affine Group

A code is said to be propelinear if its automorphism group contains a subgroup acting on its codewords regularly. A subgroup of the group \(GA(r,q)\) of affine transformations is said to be regular if it acts regularly on vectors of \(\mathbb{F}_q^r\). Every automorphism of a regular subgroup of the general affine group \(GA(r,q)\) induces a permutation on the cosets of the Hamming code of length \(\frac{q^r-1}{q-1}\). Based on this permutation, we propose a construction of \(q\)⁠-⁠ary propelinear perfect codes of length \(\frac{q^{r+1}-1}{q-1}\). In particular, for any prime \(q\) we obtain an infinite series of almost full rank \(q\)⁠-⁠ary propelinear perfect codes.

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来源期刊

Problems of Information Transmission 工程技术-计算机：理论方法

CiteScore

2.00

自引率

25.00%

发文量

审稿时长

>12 weeks

期刊介绍： Problems of Information Transmission is of interest to researcher in all fields concerned with the research and development of communication systems. This quarterly journal features coverage of statistical information theory; coding theory and techniques; noisy channels; error detection and correction; signal detection, extraction, and analysis; analysis of communication networks; optimal processing and routing; the theory of random processes; and bionics.