{"title":"Two new constructions of cyclic subspace codes via Sidon spaces","authors":"Shuhui Yu, Lijun Ji","doi":"10.1007/s10623-024-01466-y","DOIUrl":null,"url":null,"abstract":"<p>A subspace of a finite field is called a Sidon space if the product of any two of its nonzero elements is unique up to a scalar multiplier from the base field. Sidon spaces, introduced by Roth et al. in (IEEE Trans Inf Theory 64(6):4412–4422, 2018), have a close connection with optimal full-length orbit codes. In this paper, we will construct several families of large cyclic subspace codes based on the two kinds of Sidon spaces. These new codes have more codewords than the previous constructions in the literature without reducing minimum distance. In particular, in the case of <span>\\(n=4k\\)</span>, the size of our resulting code is within a factor of <span>\\(\\frac{1}{2}+o_{k}(1)\\)</span> of the sphere-packing bound as <i>k</i> goes to infinity.</p>","PeriodicalId":11130,"journal":{"name":"Designs, Codes and Cryptography","volume":"65 1","pages":""},"PeriodicalIF":1.2000,"publicationDate":"2024-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Designs, Codes and Cryptography","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10623-024-01466-y","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
引用次数: 0
Abstract
A subspace of a finite field is called a Sidon space if the product of any two of its nonzero elements is unique up to a scalar multiplier from the base field. Sidon spaces, introduced by Roth et al. in (IEEE Trans Inf Theory 64(6):4412–4422, 2018), have a close connection with optimal full-length orbit codes. In this paper, we will construct several families of large cyclic subspace codes based on the two kinds of Sidon spaces. These new codes have more codewords than the previous constructions in the literature without reducing minimum distance. In particular, in the case of \(n=4k\), the size of our resulting code is within a factor of \(\frac{1}{2}+o_{k}(1)\) of the sphere-packing bound as k goes to infinity.
如果有限域的任意两个非零元素的乘积是唯一的，直到来自基域的标量乘数，那么这个有限域的子空间就叫做西顿空间。Roth 等人在（IEEE Trans Inf Theory 64(6):4412-4422, 2018）中提出的西顿空间与最优全长轨道编码有着密切联系。本文将在这两种西顿空间的基础上，构建几族大循环子空间码。与以往文献中的构造相比，这些新编码在不减少最小距离的情况下拥有更多的码字。特别是，在 \(n=4k\) 的情况下，当 k 变为无穷大时，我们所得到的编码的大小与球形打包约束的系数在 \(\frac{1}{2}+o_{k}(1)\) 之间。
期刊介绍：
Designs, Codes and Cryptography is an archival peer-reviewed technical journal publishing original research papers in the designated areas. There is a great deal of activity in design theory, coding theory and cryptography, including a substantial amount of research which brings together more than one of the subjects. While many journals exist for each of the individual areas, few encourage the interaction of the disciplines.
The journal was founded to meet the needs of mathematicians, engineers and computer scientists working in these areas, whose interests extend beyond the bounds of any one of the individual disciplines. The journal provides a forum for high quality research in its three areas, with papers touching more than one of the areas especially welcome.
The journal also considers high quality submissions in the closely related areas of finite fields and finite geometries, which provide important tools for both the construction and the actual application of designs, codes and cryptographic systems. In particular, it includes (mostly theoretical) papers on computational aspects of finite fields. It also considers topics in sequence design, which frequently admit equivalent formulations in the journal’s main areas.
Designs, Codes and Cryptography is mathematically oriented, emphasizing the algebraic and geometric aspects of the areas it covers. The journal considers high quality papers of both a theoretical and a practical nature, provided they contain a substantial amount of mathematics.
京公网安备 11010802042870号
京ICP备2023020795号-1
分享
求助内容：
应助结果提醒方式：
扫码关注我们
