{"title":"The extended codes of some linear codes","authors":"Zhonghua Sun , Cunsheng Ding , Tingfang Chen","doi":"10.1016/j.ffa.2024.102401","DOIUrl":null,"url":null,"abstract":"<div><p>The classical way of extending an <span><math><mo>[</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>d</mi><mo>]</mo></math></span> linear code <span><math><mi>C</mi></math></span> is to add an overall parity-check coordinate to each codeword of the linear code <span><math><mi>C</mi></math></span>. This extended code, denoted by <span><math><mover><mrow><mi>C</mi></mrow><mo>‾</mo></mover><mo>(</mo><mo>−</mo><mn>1</mn><mo>)</mo></math></span> and called the standardly extended code of <span><math><mi>C</mi></math></span>, is a linear code with parameters <span><math><mo>[</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>k</mi><mo>,</mo><mover><mrow><mi>d</mi></mrow><mrow><mo>¯</mo></mrow></mover><mo>]</mo></math></span>, where <span><math><mover><mrow><mi>d</mi></mrow><mrow><mo>¯</mo></mrow></mover><mo>=</mo><mi>d</mi></math></span> or <span><math><mover><mrow><mi>d</mi></mrow><mrow><mo>¯</mo></mrow></mover><mo>=</mo><mi>d</mi><mo>+</mo><mn>1</mn></math></span>. This is one of the two extending techniques for linear codes in the literature. The standardly extended codes of some families of binary linear codes have been studied to some extent. However, not much is known about the standardly extended codes of nonbinary codes. For example, the minimum distances of the standardly extended codes of the nonbinary Hamming codes remain open for over 70 years. The first objective of this paper is to introduce the nonstandardly extended codes of a linear code and develop some general theory for this type of extended linear codes. The second objective is to study this type of extended codes of a number of families of linear codes, including cyclic codes and nonbinary Hamming codes. Four families of distance-optimal or dimension-optimal linear codes are obtained with this extending technique. The parameters of certain extended codes of many families of linear codes are settled in this paper.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"96 ","pages":"Article 102401"},"PeriodicalIF":1.2000,"publicationDate":"2024-03-12","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/S1071579724000406","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
The classical way of extending an linear code is to add an overall parity-check coordinate to each codeword of the linear code . This extended code, denoted by and called the standardly extended code of , is a linear code with parameters , where or . This is one of the two extending techniques for linear codes in the literature. The standardly extended codes of some families of binary linear codes have been studied to some extent. However, not much is known about the standardly extended codes of nonbinary codes. For example, the minimum distances of the standardly extended codes of the nonbinary Hamming codes remain open for over 70 years. The first objective of this paper is to introduce the nonstandardly extended codes of a linear code and develop some general theory for this type of extended linear codes. The second objective is to study this type of extended codes of a number of families of linear codes, including cyclic codes and nonbinary Hamming codes. Four families of distance-optimal or dimension-optimal linear codes are obtained with this extending technique. The parameters of certain extended codes of many families of linear codes are settled in this paper.
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.