{"title":"Introducing locality in some generalized AG codes","authors":"Bastien Pacifico","doi":"10.1007/s10623-025-01597-w","DOIUrl":null,"url":null,"abstract":"<p>In 1999, Xing, Niederreiter and Lam introduced a generalization of AG codes (GAG codes) using the evaluation at non-rational places of a function field. In this paper, we show that one can obtain a locality parameter <i>r</i> in such codes by using only non-rational places of degree at most <i>r</i>. This is, up to the author’s knowledge, a new way to construct locally recoverable codes (LRCs). We give an example of such a code reaching the Singleton-like bound for LRCs, and show the parameters obtained for some longer codes over <span>\\(\\mathbb F_3\\)</span>. We then investigate similarities with some concatenated codes. Contrary to previous methods, our construction allows one to obtain directly codes whose dimension is not a multiple of the locality. Finally, we give an asymptotic study using the Garcia–Stichtenoth tower of function fields, for both our construction with GAG codes and a construction of concatenated codes. We give explicit infinite families of LRCs with locality 2 over any finite field of cardinality greater than 3 following our approach with GAG codes.</p>","PeriodicalId":11130,"journal":{"name":"Designs, Codes and Cryptography","volume":"24 1","pages":""},"PeriodicalIF":1.4000,"publicationDate":"2025-02-24","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/s10623-025-01597-w","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
In 1999, Xing, Niederreiter and Lam introduced a generalization of AG codes (GAG codes) using the evaluation at non-rational places of a function field. In this paper, we show that one can obtain a locality parameter r in such codes by using only non-rational places of degree at most r. This is, up to the author’s knowledge, a new way to construct locally recoverable codes (LRCs). We give an example of such a code reaching the Singleton-like bound for LRCs, and show the parameters obtained for some longer codes over \(\mathbb F_3\). We then investigate similarities with some concatenated codes. Contrary to previous methods, our construction allows one to obtain directly codes whose dimension is not a multiple of the locality. Finally, we give an asymptotic study using the Garcia–Stichtenoth tower of function fields, for both our construction with GAG codes and a construction of concatenated codes. We give explicit infinite families of LRCs with locality 2 over any finite field of cardinality greater than 3 following our approach with GAG 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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