Alexander Barg, Ohad Elishco, Ryan Gabrys, Geyang Wang, Eitan Yaakobi
{"title":"线路和电网上的存储代码和可恢复系统","authors":"Alexander Barg, Ohad Elishco, Ryan Gabrys, Geyang Wang, Eitan Yaakobi","doi":"10.1007/s10623-024-01481-z","DOIUrl":null,"url":null,"abstract":"<p>A storage code is an assignment of symbols to the vertices of a connected graph <i>G</i>(<i>V</i>, <i>E</i>) with the property that the value of each vertex is a function of the values of its neighbors, or more generally, of a certain neighborhood of the vertex in <i>G</i>. In this work we introduce a new construction method of storage codes, enabling one to construct new codes from known ones via an interleaving procedure driven by resolvable designs. We also study storage codes on <span>\\({\\mathbb Z}\\)</span> and <span>\\({\\mathbb Z}^2\\)</span> (lines and grids), finding closed-form expressions for the capacity of several one and two-dimensional systems depending on their recovery set, using connections between storage codes, graphs, anticodes, and difference-avoiding sets.</p>","PeriodicalId":11130,"journal":{"name":"Designs, Codes and Cryptography","volume":"9 1","pages":""},"PeriodicalIF":1.4000,"publicationDate":"2024-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Storage codes and recoverable systems on lines and grids\",\"authors\":\"Alexander Barg, Ohad Elishco, Ryan Gabrys, Geyang Wang, Eitan Yaakobi\",\"doi\":\"10.1007/s10623-024-01481-z\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>A storage code is an assignment of symbols to the vertices of a connected graph <i>G</i>(<i>V</i>, <i>E</i>) with the property that the value of each vertex is a function of the values of its neighbors, or more generally, of a certain neighborhood of the vertex in <i>G</i>. In this work we introduce a new construction method of storage codes, enabling one to construct new codes from known ones via an interleaving procedure driven by resolvable designs. We also study storage codes on <span>\\\\({\\\\mathbb Z}\\\\)</span> and <span>\\\\({\\\\mathbb Z}^2\\\\)</span> (lines and grids), finding closed-form expressions for the capacity of several one and two-dimensional systems depending on their recovery set, using connections between storage codes, graphs, anticodes, and difference-avoiding sets.</p>\",\"PeriodicalId\":11130,\"journal\":{\"name\":\"Designs, Codes and Cryptography\",\"volume\":\"9 1\",\"pages\":\"\"},\"PeriodicalIF\":1.4000,\"publicationDate\":\"2024-09-01\",\"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/s10623-024-01481-z\",\"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/s10623-024-01481-z","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
摘要
存储代码是对连通图 G（V，E）顶点的符号赋值，其特性是每个顶点的值都是其邻域值的函数，或者更广泛地说，是顶点在 G 中的某个邻域值的函数。在这项工作中，我们引入了一种新的存储代码构造方法，通过可解析设计驱动的交织程序，人们可以从已知代码中构造出新的代码。我们还研究了\({\mathbb Z}\)和\({\mathbb Z}^2\)（线和网格）上的存储编码，利用存储编码、图、反编码和避差集之间的联系，找到了几种一维和二维系统容量的闭式表达式，这取决于它们的恢复集。
Storage codes and recoverable systems on lines and grids
A storage code is an assignment of symbols to the vertices of a connected graph G(V, E) with the property that the value of each vertex is a function of the values of its neighbors, or more generally, of a certain neighborhood of the vertex in G. In this work we introduce a new construction method of storage codes, enabling one to construct new codes from known ones via an interleaving procedure driven by resolvable designs. We also study storage codes on \({\mathbb Z}\) and \({\mathbb Z}^2\) (lines and grids), finding closed-form expressions for the capacity of several one and two-dimensional systems depending on their recovery set, using connections between storage codes, graphs, anticodes, and difference-avoiding sets.
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.
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号
