Finite Fields and Their Applications最新文献_第7页

The second generalized covering radius of binary primitive double-error-correcting BCH codes 二元基元双纠错BCH码的第二广义覆盖半径

IF 1.2 3区数学

Finite Fields and Their Applications Pub Date : 2025-05-07 DOI: 10.1016/j.ffa.2025.102648

Lev Yohananov , Moshe Schwartz

引用次数: 0

On the b-symbol weights of linear codes for large b 大b线性码的b符号权值

IF 1.2 3区数学

Finite Fields and Their Applications Pub Date : 2025-05-07 DOI: 10.1016/j.ffa.2025.102647

Dongmei Huang , Qunying Liao , Gaohua Tang , Shixin Zhu

{"title":"On the b-symbol weights of linear codes for large b","authors":"Dongmei Huang , Qunying Liao , Gaohua Tang , Shixin Zhu","doi":"10.1016/j.ffa.2025.102647","DOIUrl":"10.1016/j.ffa.2025.102647","url":null,"abstract":"<div><div>Recently, b-symbol metric linear codes, which are extensions of linear codes with Hamming weight and symbol-pair weight, have attracted much attention due to their repair efficiency in b-symbol read channels. Extensive efforts have been dedicated to studying the classic Hamming weight and symbol-pair weight of linear codes, but there has been relatively less progress regarding the general b-symbol weight for <math><mi>b</mi><mo>≥</mo><mn>3</mn></math>. In this paper, we mainly consider the b-symbol weight of linear codes for large b. Firstly, we establish a direct correspondence between the b-symbol weight distribution of a linear code and the Hamming weight distribution of another linear code. Using this relation, we derive a Griesmer-type bound on the minimum b-symbol weight, which partially settles a conjecture of Shi, Zhu and Helleseth. Furthermore, we discover that the minimum b-symbol distance of a <math><mo>[</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>]</mo></math> cyclic code is equal to the minimum b-symbol distance of certain linear code of parameters <math><mo>[</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>−</mo><mi>b</mi><mo>]</mo></math> and prove that every <math><mo>[</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>]</mo></math> cyclic code is always b-symbol MDS for <math><mi>b</mi><mo>≥</mo><mi>min</mi><mo>⁡</mo><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mi>k</mi><mo>,</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>−</mo><mrow><mo>⌈</mo><mfrac><mrow><msup><mrow><mi>d</mi></mrow><mrow><mo>⊥</mo></mrow></msup></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌉</mo></mrow><mo>)</mo></mrow></math>, where <math><msup><mrow><mi>d</mi></mrow><mrow><mo>⊥</mo></mrow></msup></math> is the minimum distance of its dual. When <math><mi>b</mi><mo>≥</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>−</mo><mrow><mo>⌈</mo><mfrac><mrow><msup><mrow><mi>d</mi></mrow><mrow><mo>⊥</mo></mrow></msup></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌉</mo></mrow></math> we also determine its b-symbol weight distribution. Finally, we investigate the minimum b-symbol weight of several specific cyclic codes, including QR codes, irreducible cyclic codes, Kasami codes and the duals of double-error-correcting BCH codes.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"107 ","pages":"Article 102647"},"PeriodicalIF":1.2,"publicationDate":"2025-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143918441","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Some counterexamples for a version of Springer's theorem on forms of higher degree 关于高次形式的施普林格定理的一些反例

IF 1.2 3区数学

Finite Fields and Their Applications Pub Date : 2025-05-07 DOI: 10.1016/j.ffa.2025.102646

Alexander S. Sivatski

{"title":"Some counterexamples for a version of Springer's theorem on forms of higher degree","authors":"Alexander S. Sivatski","doi":"10.1016/j.ffa.2025.102646","DOIUrl":"10.1016/j.ffa.2025.102646","url":null,"abstract":"<div><div>Let φ be a form (homogeneous polynomial) of degree d in <math><mi>n</mi><mo>≥</mo><mn>2</mn></math> variables over a field F. We call φ anisotropic over F, if the equality <math><mi>φ</mi><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>=</mo><mn>0</mn></math> with <math><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∈</mo><mi>F</mi></math> implies <math><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mo>…</mo><mo>=</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mn>0</mn></math>. Otherwise φ is called isotropic. Assume that <math><mi>L</mi><mo>/</mo><mi>F</mi></math> be a finite field extension, the numbers d and <math><mo>[</mo><mi>L</mi><mo>:</mo><mi>F</mi><mo>]</mo></math> are coprime, and the form φ is anisotropic and diagonal. Does the form φ remain anisotropic over L? This problem can be considered as an analog of the Springer theorem on behaviour of anisotropic quadratic forms under odd degree extensions. In particular, we investigate this problem in the case <math><mi>F</mi><mo>=</mo><mi>Q</mi></math>. We give examples of extensions of degree 2, 3, and 5, which show that in general the answer is negative and pose a few related questions. Cubic extensions are treated in the arithmetic as well as in the general case. Finite fields <math><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></math> and their extensions is the principal tool in constructing the counterexamples in question.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"107 ","pages":"Article 102646"},"PeriodicalIF":1.2,"publicationDate":"2025-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143918443","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

The automorphism group of the pn-torsion points of an elliptic curve over a field of characteristic p ≥ 5 特征为p ≥ 的场上椭圆曲线pn-扭转点的自同构群

IF 1.2 3区数学

Finite Fields and Their Applications Pub Date : 2025-04-22 DOI: 10.1016/j.ffa.2025.102631

Bo-Hae Im , Hansol Kim

引用次数: 0

(σ,τ)-derivations of group rings with applications （σ,τ）-群环的导数及其应用

IF 1.2 3区数学

Finite Fields and Their Applications Pub Date : 2025-04-22 DOI: 10.1016/j.ffa.2025.102629

Praveen Manju , Rajendra Kumar Sharma

引用次数: 0

The minimum distance of a parameterized code over an even cycle 参数化代码在偶数周期上的最小距离

IF 1.2 3区数学

Finite Fields and Their Applications Pub Date : 2025-04-15 DOI: 10.1016/j.ffa.2025.102630

Eduardo Camps-Moreno , Jorge Neves , Eliseo Sarmiento

引用次数: 0

Stephen D. Cohen's contributions to the finite fields community Stephen D. Cohen对有限域界的贡献

IF 1.2 3区数学

Finite Fields and Their Applications Pub Date : 2025-04-10 DOI: 10.1016/j.ffa.2025.102627

Sophie Huczynska, Gary L. Mullen

引用次数: 0

Cyclically covering subspaces of Fqn 循环覆盖Fqn的子空间

IF 1.2 3区数学

Finite Fields and Their Applications Pub Date : 2025-04-10 DOI: 10.1016/j.ffa.2025.102625

Meng Sun, Changli Ma, Liwei Zeng

引用次数: 0

The study on three classes of permutation quadrinomials over finite fields of odd characteristic 奇特征有限域上三类置换四项的研究

IF 1.2 3区数学

Finite Fields and Their Applications Pub Date : 2025-04-10 DOI: 10.1016/j.ffa.2025.102626

Changhui Chen , Haibin Kan , Jie Peng , Hengtai Wang , Lijing Zheng

引用次数: 0

Some classes of permutation pentanomials 几类置换五反常项

IF 1.2 3区数学

Finite Fields and Their Applications Pub Date : 2025-03-27 DOI: 10.1016/j.ffa.2025.102619

Zhiguo Ding , Michael E. Zieve

{"title":"Some classes of permutation pentanomials","authors":"Zhiguo Ding , Michael E. Zieve","doi":"10.1016/j.ffa.2025.102619","DOIUrl":"10.1016/j.ffa.2025.102619","url":null,"abstract":"<div><div>For each prime <math><mi>p</mi><mo>≠</mo><mn>3</mn></math> and each power <math><mi>q</mi><mo>=</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>k</mi></mrow></msup></math>, we present two large classes of permutation polynomials over <math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub></math> of the form <math><msup><mrow><mi>X</mi></mrow><mrow><mi>r</mi></mrow></msup><mi>B</mi><mo>(</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>q</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></math> which have at most five terms, where <math><mi>B</mi><mo>(</mo><mi>X</mi><mo>)</mo></math> is a polynomial with coefficients in <math><mo>{</mo><mn>1</mn><mo>,</mo><mo>−</mo><mn>1</mn><mo>}</mo></math>. The special case <math><mi>p</mi><mo>=</mo><mn>2</mn></math> of our results comprises a vast generalization of 76 recent results and conjectures in the literature. In case <math><mi>p</mi><mo>></mo><mn>2</mn></math>, no instances of our permutation polynomials have appeared in the literature, and the construction of such polynomials had been posed as an open problem. Our proofs are short and involve no computations, in contrast to the proofs of many of the special cases of our results which were published previously.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"106 ","pages":"Article 102619"},"PeriodicalIF":1.2,"publicationDate":"2025-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143705967","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0