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

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

Abstract

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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基于一般仿射群正则子群的q元线性完美码
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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来源期刊
Problems of Information Transmission
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
10
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.
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