{"title":"New and improved formally self-dual codes with small hulls from polynomial four Toeplitz codes","authors":"Yang Li, Shitao Li, Shixin Zhu","doi":"10.1007/s10623-024-01460-4","DOIUrl":null,"url":null,"abstract":"<p>Formally self-dual (FSD) codes and linear codes with small Euclidean (resp. Hermitian) hulls have recently attracted a lot of attention due to their theoretical and practical importance. However, there has been not much attention on FSD codes with small hulls. In this paper, we introduce two kinds of polynomial four Toeplitz codes and prove that they must be FSD. We characterize the linear complementary dual (LCD) properties and one-dimensional hull properties of such codes with respect to the Euclidean and Hermitian inner products. Using these characterizations, we find some improved binary, ternary Euclidean and quaternary Hermitian FSD LCD codes, as well as many non-equivalent ones that perform equally well with respect to best-known (FSD) LCD codes in the literature. Furthermore, some (near) maximum distance separable FSD codes with both one-dimensional Euclidean hull and one-dimensional Hermitian hull are also given as examples.</p>","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10623-024-01460-4","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0
Abstract
Formally self-dual (FSD) codes and linear codes with small Euclidean (resp. Hermitian) hulls have recently attracted a lot of attention due to their theoretical and practical importance. However, there has been not much attention on FSD codes with small hulls. In this paper, we introduce two kinds of polynomial four Toeplitz codes and prove that they must be FSD. We characterize the linear complementary dual (LCD) properties and one-dimensional hull properties of such codes with respect to the Euclidean and Hermitian inner products. Using these characterizations, we find some improved binary, ternary Euclidean and quaternary Hermitian FSD LCD codes, as well as many non-equivalent ones that perform equally well with respect to best-known (FSD) LCD codes in the literature. Furthermore, some (near) maximum distance separable FSD codes with both one-dimensional Euclidean hull and one-dimensional Hermitian hull are also given as examples.
期刊介绍：
Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance.
Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.
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