{"title":"New constructions of optimal (r,δ)-LRCs via good polynomials","authors":"Yuan Gao, Siman Yang","doi":"10.1016/j.ffa.2024.102362","DOIUrl":null,"url":null,"abstract":"<div><p>Locally repairable codes (LRCs) are a class of erasure codes that are widely used in distributed storage systems, which allow for efficient recovery of data in the case of node failures or data loss. In 2014, Tamo and Barg introduced Reed-Solomon-like (RS-like) Singleton-optimal <span><math><mo>(</mo><mi>r</mi><mo>,</mo><mi>δ</mi><mo>)</mo></math></span><span>-LRCs based on polynomial evaluation<span>. These constructions rely on the existence of so-called good polynomial that is constant on each of some pairwise disjoint subsets of </span></span><span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>. In this paper, we extend the aforementioned constructions of RS-like LRCs and propose new constructions of <span><math><mo>(</mo><mi>r</mi><mo>,</mo><mi>δ</mi><mo>)</mo></math></span>-LRCs whose code length can be larger. These new <span><math><mo>(</mo><mi>r</mi><mo>,</mo><mi>δ</mi><mo>)</mo></math></span>-LRCs are all distance-optimal, namely, they attain an upper bound on the minimum distance that will be established in this paper. This bound is sharper than the Singleton-type bound in some cases owing to the extra conditions, it coincides with the Singleton-type bound for certain cases. Combining our constructions with known explicit good polynomials of special forms, we can get various explicit Singleton-optimal <span><math><mo>(</mo><mi>r</mi><mo>,</mo><mi>δ</mi><mo>)</mo></math></span>-LRCs with new parameters, whose code lengths are all larger than that constructed by the RS-like <span><math><mo>(</mo><mi>r</mi><mo>,</mo><mi>δ</mi><mo>)</mo></math></span>-LRCs introduced by Tamo and Barg. Note that the code length of classical RS codes and RS-like LRCs are both bounded by the field size. We explicitly construct the Singleton-optimal <span><math><mo>(</mo><mi>r</mi><mo>,</mo><mi>δ</mi><mo>)</mo></math></span>-LRCs with length <span><math><mi>n</mi><mo>=</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo>+</mo><mi>δ</mi></math></span> for any positive integers <span><math><mi>r</mi><mo>,</mo><mi>δ</mi><mo>≥</mo><mn>2</mn></math></span> and <span><math><mo>(</mo><mi>r</mi><mo>+</mo><mi>δ</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>|</mo><mo>(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span>. We also show the existence of Singleton-optimal <span><math><mo>(</mo><mi>r</mi><mo>,</mo><mi>δ</mi><mo>)</mo></math></span>-LRCs with length <span><math><mi>q</mi><mo>+</mo><mi>δ</mi></math></span> over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>=</mo><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>a</mi></mrow></msup></mrow></msub></math></span> (<span><math><mi>a</mi><mo>≥</mo><mn>3</mn></math></span>) provided <span><math><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>|</mo><mo>(</mo><mi>r</mi><mo>+</mo><mi>δ</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span>, <span><math><mo>(</mo><mi>r</mi><mo>+</mo><mi>δ</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>|</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>a</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span> and <span><math><mi>p</mi><mo>|</mo><mi>δ</mi></math></span>. When <em>δ</em> is proportional to <em>q</em><span>, they are asymptotically longer than that constructed via elliptic curves whose length is at most </span><span><math><mi>q</mi><mo>+</mo><mn>2</mn><msqrt><mrow><mi>q</mi></mrow></msqrt></math></span>. Besides, they allow more flexibility on the values of <em>r</em> and <em>δ</em>.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":null,"pages":null},"PeriodicalIF":1.2000,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Finite Fields and Their Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S1071579724000029","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Locally repairable codes (LRCs) are a class of erasure codes that are widely used in distributed storage systems, which allow for efficient recovery of data in the case of node failures or data loss. In 2014, Tamo and Barg introduced Reed-Solomon-like (RS-like) Singleton-optimal -LRCs based on polynomial evaluation. These constructions rely on the existence of so-called good polynomial that is constant on each of some pairwise disjoint subsets of . In this paper, we extend the aforementioned constructions of RS-like LRCs and propose new constructions of -LRCs whose code length can be larger. These new -LRCs are all distance-optimal, namely, they attain an upper bound on the minimum distance that will be established in this paper. This bound is sharper than the Singleton-type bound in some cases owing to the extra conditions, it coincides with the Singleton-type bound for certain cases. Combining our constructions with known explicit good polynomials of special forms, we can get various explicit Singleton-optimal -LRCs with new parameters, whose code lengths are all larger than that constructed by the RS-like -LRCs introduced by Tamo and Barg. Note that the code length of classical RS codes and RS-like LRCs are both bounded by the field size. We explicitly construct the Singleton-optimal -LRCs with length for any positive integers and . We also show the existence of Singleton-optimal -LRCs with length over () provided , and . When δ is proportional to q, they are asymptotically longer than that constructed via elliptic curves whose length is at most . Besides, they allow more flexibility on the values of r and δ.
期刊介绍:
Finite Fields and Their Applications is a peer-reviewed technical journal publishing papers in finite field theory as well as in applications of finite fields. As a result of applications in a wide variety of areas, finite fields are increasingly important in several areas of mathematics, including linear and abstract algebra, number theory and algebraic geometry, as well as in computer science, statistics, information theory, and engineering.
For cohesion, and because so many applications rely on various theoretical properties of finite fields, it is essential that there be a core of high-quality papers on theoretical aspects. In addition, since much of the vitality of the area comes from computational problems, the journal publishes papers on computational aspects of finite fields as well as on algorithms and complexity of finite field-related methods.
The journal also publishes papers in various applications including, but not limited to, algebraic coding theory, cryptology, combinatorial design theory, pseudorandom number generation, and linear recurring sequences. There are other areas of application to be included, but the important point is that finite fields play a nontrivial role in the theory, application, or algorithm.