{"title":"On finding the largest minimum distance of locally recoverable codes: A graph theory approach","authors":"Majid Khabbazian , Muriel Médard","doi":"10.1016/j.disc.2024.114298","DOIUrl":null,"url":null,"abstract":"<div><div>The <span><math><mo>[</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>r</mi><mo>]</mo></math></span>-Locally recoverable codes (LRC) studied in this work are a well-studied family of <span><math><mo>[</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>]</mo></math></span> linear codes for which the value of each symbol can be recovered by a linear combination of at most <em>r</em> other symbols. In this paper, we study the <em>LMD</em> problem, which is to find the largest possible minimum distance of <span><math><mo>[</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>r</mi><mo>]</mo></math></span>-LRCs, denoted by <span><math><mi>D</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>r</mi><mo>)</mo></math></span>. LMD can be approximated within an additive term of one—it is known that <span><math><mi>D</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>r</mi><mo>)</mo></math></span> is equal to either <span><math><msup><mrow><mi>d</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> or <span><math><msup><mrow><mi>d</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>−</mo><mn>1</mn></math></span>, where <span><math><msup><mrow><mi>d</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>=</mo><mi>n</mi><mo>−</mo><mi>k</mi><mo>−</mo><mrow><mo>⌈</mo><mfrac><mrow><mi>k</mi></mrow><mrow><mi>r</mi></mrow></mfrac><mo>⌉</mo></mrow><mo>+</mo><mn>2</mn></math></span>. Moreover, for a range of parameters, it is known precisely whether the distance <span><math><mi>D</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>r</mi><mo>)</mo></math></span> is <span><math><msup><mrow><mi>d</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> or <span><math><msup><mrow><mi>d</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>−</mo><mn>1</mn></math></span>. However, the problem is still open despite a significant effort. In this work, we convert LMD to an equivalent simply-stated problem in graph theory. Using this conversion, we show that an instance of LMD is at least as hard as computing the size of a maximal graph of high girth, a hard problem in extremal graph theory. This is an evidence that LMD—although can be approximated within an additive term of one—is hard to solve in general. As a positive result, thanks to the conversion and the exiting results in extremal graph theory, we solve LMD for a range of code parameters that has not been solved before.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 2","pages":"Article 114298"},"PeriodicalIF":0.7000,"publicationDate":"2024-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0012365X24004291","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
The -Locally recoverable codes (LRC) studied in this work are a well-studied family of linear codes for which the value of each symbol can be recovered by a linear combination of at most r other symbols. In this paper, we study the LMD problem, which is to find the largest possible minimum distance of -LRCs, denoted by . LMD can be approximated within an additive term of one—it is known that is equal to either or , where . Moreover, for a range of parameters, it is known precisely whether the distance is or . However, the problem is still open despite a significant effort. In this work, we convert LMD to an equivalent simply-stated problem in graph theory. Using this conversion, we show that an instance of LMD is at least as hard as computing the size of a maximal graph of high girth, a hard problem in extremal graph theory. This is an evidence that LMD—although can be approximated within an additive term of one—is hard to solve in general. As a positive result, thanks to the conversion and the exiting results in extremal graph theory, we solve LMD for a range of code parameters that has not been solved before.
期刊介绍:
Discrete Mathematics provides a common forum for significant research in many areas of discrete mathematics and combinatorics. Among the fields covered by Discrete Mathematics are graph and hypergraph theory, enumeration, coding theory, block designs, the combinatorics of partially ordered sets, extremal set theory, matroid theory, algebraic combinatorics, discrete geometry, matrices, and discrete probability theory.
Items in the journal include research articles (Contributions or Notes, depending on length) and survey/expository articles (Perspectives). Efforts are made to process the submission of Notes (short articles) quickly. The Perspectives section features expository articles accessible to a broad audience that cast new light or present unifying points of view on well-known or insufficiently-known topics.