{"title":"加性扭曲码：新的距离边界和量子码的无限族","authors":"Reza Dastbasteh, Petr Lisoněk","doi":"10.1007/s10623-024-01552-1","DOIUrl":null,"url":null,"abstract":"<p>We provide a new construction of quantum codes that enables integration of a broader class of classical codes into the mathematical framework of quantum stabilizer codes. Next, we present new connections between twisted codes and linear cyclic codes and provide novel bounds for the minimum distance of twisted codes. We show that classical tools such as the Hartmann–Tzeng minimum distance bound are applicable to twisted codes. This enabled us to discover five new infinite families and many other examples of record-breaking, and sometimes optimal, binary quantum codes. We also discuss the role of the <span>\\(\\gamma \\)</span> value on the parameters of twisted codes and present new results regarding the construction of twisted codes with different <span>\\(\\gamma \\)</span> values but identical parameters. Finally, we list many new record-breaking binary quantum codes that we obtained from additive twisted, linear cyclic, and constacyclic codes.</p>","PeriodicalId":11130,"journal":{"name":"Designs, Codes and Cryptography","volume":"77 1","pages":""},"PeriodicalIF":1.4000,"publicationDate":"2025-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Additive twisted codes: new distance bounds and infinite families of quantum codes\",\"authors\":\"Reza Dastbasteh, Petr Lisoněk\",\"doi\":\"10.1007/s10623-024-01552-1\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We provide a new construction of quantum codes that enables integration of a broader class of classical codes into the mathematical framework of quantum stabilizer codes. Next, we present new connections between twisted codes and linear cyclic codes and provide novel bounds for the minimum distance of twisted codes. We show that classical tools such as the Hartmann–Tzeng minimum distance bound are applicable to twisted codes. This enabled us to discover five new infinite families and many other examples of record-breaking, and sometimes optimal, binary quantum codes. We also discuss the role of the <span>\\\\(\\\\gamma \\\\)</span> value on the parameters of twisted codes and present new results regarding the construction of twisted codes with different <span>\\\\(\\\\gamma \\\\)</span> values but identical parameters. Finally, we list many new record-breaking binary quantum codes that we obtained from additive twisted, linear cyclic, and constacyclic codes.</p>\",\"PeriodicalId\":11130,\"journal\":{\"name\":\"Designs, Codes and Cryptography\",\"volume\":\"77 1\",\"pages\":\"\"},\"PeriodicalIF\":1.4000,\"publicationDate\":\"2025-01-16\",\"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/s10623-024-01552-1\",\"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}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Designs, Codes and Cryptography","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10623-024-01552-1","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}
Additive twisted codes: new distance bounds and infinite families of quantum codes
We provide a new construction of quantum codes that enables integration of a broader class of classical codes into the mathematical framework of quantum stabilizer codes. Next, we present new connections between twisted codes and linear cyclic codes and provide novel bounds for the minimum distance of twisted codes. We show that classical tools such as the Hartmann–Tzeng minimum distance bound are applicable to twisted codes. This enabled us to discover five new infinite families and many other examples of record-breaking, and sometimes optimal, binary quantum codes. We also discuss the role of the \(\gamma \) value on the parameters of twisted codes and present new results regarding the construction of twisted codes with different \(\gamma \) values but identical parameters. Finally, we list many new record-breaking binary quantum codes that we obtained from additive twisted, linear cyclic, and constacyclic codes.
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.
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