PG(2,q4) 中的饱和线性集合

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications Pub Date : 2024-05-21 DOI:10.1016/j.ffa.2024.102447

Ferdinando Zullo

{"title":"PG(2,q4) 中的饱和线性集合","authors":"Ferdinando Zullo","doi":"10.1016/j.ffa.2024.102447","DOIUrl":null,"url":null,"abstract":"<div>Bonini, Borello and Byrne started the study of saturating linear sets in Desarguesian projective spaces, in connection with the covering problem in the rank metric. In this paper we study 1-saturating linear sets in PG<math><mo>(</mo><mn>2</mn><mo>,</mo><msup><mrow><mi>q</mi></mrow><mrow><mn>4</mn></mrow></msup><mo>)</mo></math>, that is <math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math>-linear sets in PG<math><mo>(</mo><mn>2</mn><mo>,</mo><msup><mrow><mi>q</mi></mrow><mrow><mn>4</mn></mrow></msup><mo>)</mo></math> with the property that their secant lines cover the entire plane. By making use of a characterization of generalized Gabidulin codes, we prove that the rank of such a linear set is at least 5. This answers to a recent question posed by Bartoli, Borello and Marino.</div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":null,"pages":null},"PeriodicalIF":1.2000,"publicationDate":"2024-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S1071579724000868/pdfft?md5=8d9b69a18531ce3d150444da9e0bf2f2&pid=1-s2.0-S1071579724000868-main.pdf","citationCount":"0","resultStr":"{\"title\":\"Saturating linear sets in PG(2,q4)\",\"authors\":\"Ferdinando Zullo\",\"doi\":\"10.1016/j.ffa.2024.102447\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div>Bonini, Borello and Byrne started the study of saturating linear sets in Desarguesian projective spaces, in connection with the covering problem in the rank metric. In this paper we study 1-saturating linear sets in PG<math><mo>(</mo><mn>2</mn><mo>,</mo><msup><mrow><mi>q</mi></mrow><mrow><mn>4</mn></mrow></msup><mo>)</mo></math>, that is <math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math>-linear sets in PG<math><mo>(</mo><mn>2</mn><mo>,</mo><msup><mrow><mi>q</mi></mrow><mrow><mn>4</mn></mrow></msup><mo>)</mo></math> with the property that their secant lines cover the entire plane. By making use of a characterization of generalized Gabidulin codes, we prove that the rank of such a linear set is at least 5. This answers to a recent question posed by Bartoli, Borello and Marino.</div>\",\"PeriodicalId\":50446,\"journal\":{\"name\":\"Finite Fields and Their Applications\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.2000,\"publicationDate\":\"2024-05-21\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://www.sciencedirect.com/science/article/pii/S1071579724000868/pdfft?md5=8d9b69a18531ce3d150444da9e0bf2f2&pid=1-s2.0-S1071579724000868-main.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Finite Fields and Their Applications\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S1071579724000868\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Finite Fields and Their Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S1071579724000868","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

摘要

Bonini、Borello 和 Byrne 结合秩度量中的覆盖问题，开始了对德萨格投影空间中饱和线性集的研究。在本文中，我们研究了 PG(2,q4) 中的 1 饱和线性集，即 PG(2,q4) 中的 Fq 线性集，它们的正割线覆盖整个平面。通过利用广义加比杜林码的特性，我们证明了这种线性集的秩至少为 5。这回答了巴托利、博雷洛和马里诺最近提出的一个问题。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文本刊更多论文

Saturating linear sets in PG(2,q4)

Bonini, Borello and Byrne started the study of saturating linear sets in Desarguesian projective spaces, in connection with the covering problem in the rank metric. In this paper we study 1-saturating linear sets in PG $(2, q^{4})$ , that is $F_{q}$ -linear sets in PG $(2, q^{4})$ with the property that their secant lines cover the entire plane. By making use of a characterization of generalized Gabidulin codes, we prove that the rank of such a linear set is at least 5. This answers to a recent question posed by Bartoli, Borello and Marino.

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助

来源期刊

Finite Fields and Their Applications 数学-数学

CiteScore

2.00

自引率

20.00%

发文量

133

审稿时长

6-12 weeks

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.