Finite Fields and Their Applications最新文献

{"title":"The automorphism group of the pn-torsion points of an elliptic curve over a field of characteristic p ≥ 5","authors":"Bo-Hae Im ,&nbsp;Hansol Kim","doi":"10.1016/j.ffa.2025.102631","DOIUrl":"10.1016/j.ffa.2025.102631","url":null,"abstract":"<div><div>For a field <em>K</em> of characteristic <span><math><mi>p</mi><mo>≥</mo><mn>5</mn></math></span> and the elliptic curve <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msub><mo>:</mo><msup><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>+</mo><mi>s</mi><mi>x</mi><mo>+</mo><mi>t</mi></math></span> defined over the function field <span><math><mi>K</mi><mrow><mo>(</mo><mi>s</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow></math></span> of two variables <em>s</em> and <em>t</em>, we prove that for a positive integer <em>n</em>, the automorphism group of the normal extension <span><math><mi>K</mi><mrow><mo>(</mo><mi>s</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mrow><mo>(</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msub><mrow><mo>[</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>]</mo></mrow><mo>)</mo></mrow><mo>/</mo><mi>K</mi><mrow><mo>(</mo><mi>s</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow></math></span> is isomorphic to <span><math><msup><mrow><mo>(</mo><mi>Z</mi><mo>/</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup><mi>Z</mi><mo>)</mo></mrow><mrow><mo>×</mo></mrow></msup></math></span>, and its inseparable degree is <span><math><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"106 ","pages":"Article 102631"},"PeriodicalIF":1.2,"publicationDate":"2025-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143859919","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}

{"title":"(σ,τ)-derivations of group rings with applications","authors":"Praveen Manju ,&nbsp;Rajendra Kumar Sharma","doi":"10.1016/j.ffa.2025.102629","DOIUrl":"10.1016/j.ffa.2025.102629","url":null,"abstract":"&lt;div&gt;&lt;div&gt;Leo Creedon and Kieran Hughes in &lt;span&gt;&lt;span&gt;[18]&lt;/span&gt;&lt;/span&gt; studied derivations of a group ring &lt;em&gt;RG&lt;/em&gt; (of a group &lt;em&gt;G&lt;/em&gt; over a commutative unital ring &lt;em&gt;R&lt;/em&gt;) in terms of generators and relators of group &lt;em&gt;G&lt;/em&gt;. In this article, we do that for &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;σ&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;τ&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;-derivations. We develop a necessary and sufficient condition such that a map &lt;span&gt;&lt;math&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;→&lt;/mo&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; can be extended uniquely to a &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;σ&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;τ&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;-derivation &lt;em&gt;D&lt;/em&gt; of &lt;em&gt;RG&lt;/em&gt;, where &lt;em&gt;R&lt;/em&gt; is a commutative ring with unity, &lt;em&gt;G&lt;/em&gt; is a group having a presentation &lt;span&gt;&lt;math&gt;&lt;mo&gt;〈&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;〉&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; (&lt;em&gt;X&lt;/em&gt; the set of generators and &lt;em&gt;Y&lt;/em&gt; the set of relators) and &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;σ&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;τ&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; is a pair of &lt;em&gt;R&lt;/em&gt;-algebra endomorphisms of &lt;em&gt;RG&lt;/em&gt; which are &lt;em&gt;R&lt;/em&gt;-linear extensions of the group endomorphisms of &lt;em&gt;G&lt;/em&gt;. Further, we classify all inner &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;σ&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;τ&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;-derivations of the group algebra &lt;em&gt;RG&lt;/em&gt; of an arbitrary group &lt;em&gt;G&lt;/em&gt; over an arbitrary commutative unital ring &lt;em&gt;R&lt;/em&gt; in terms of the rank and a basis of the corresponding &lt;em&gt;R&lt;/em&gt;-module consisting of all inner &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;σ&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;τ&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;-derivations of &lt;em&gt;RG&lt;/em&gt;. We obtain several corollaries, particularly when &lt;em&gt;G&lt;/em&gt; is a &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;σ&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;τ&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;-FC group or a finite group &lt;em&gt;G&lt;/em&gt; and when &lt;em&gt;R&lt;/em&gt; is a field. We also prove that if &lt;em&gt;R&lt;/em&gt; is a unital ring and &lt;em&gt;G&lt;/em&gt; is a group whose order is invertible in &lt;em&gt;R&lt;/em&gt;, then every &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;σ&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;τ&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;-derivation of &lt;em&gt;RG&lt;/em&gt; is inner. We apply the results obtained above to study &lt;em&gt;σ&lt;/em&gt;-derivations of commutative group algebras over a field of positive characteristic and to classify all inner and outer &lt;em&gt;σ&lt;/em&gt;-derivations of dihedral group algebras &lt;span&gt;&lt;math&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; (&lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;〈&lt;/mo&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;〉&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;) over an arbitrary field &lt;span&gt;&lt;math&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; of any characteristic. Finally, we g","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"107 ","pages":"Article 102629"},"PeriodicalIF":1.2,"publicationDate":"2025-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143859446","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}

{"title":"The minimum distance of a parameterized code over an even cycle","authors":"Eduardo Camps-Moreno ,&nbsp;Jorge Neves ,&nbsp;Eliseo Sarmiento","doi":"10.1016/j.ffa.2025.102630","DOIUrl":"10.1016/j.ffa.2025.102630","url":null,"abstract":"<div><div>Parameterized linear codes over a graph exhibit a very interesting relation between their basic parameters and the combinatorics of the graph. We address the computation of the most elusive of all parameters, the minimum distance. We focus on the case of the parameterized code of order 1 over an even cycle.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"106 ","pages":"Article 102630"},"PeriodicalIF":1.2,"publicationDate":"2025-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143829787","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}

{"title":"Stephen D. Cohen's contributions to the finite fields community","authors":"Sophie Huczynska,&nbsp;Gary L. Mullen","doi":"10.1016/j.ffa.2025.102627","DOIUrl":"10.1016/j.ffa.2025.102627","url":null,"abstract":"<div><div>In this short note, to mark the retirement of Stephen D. Cohen from his editorial role at this journal, we discuss and acknowledge his significant contributions to the finite fields community in the areas of service and research.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"106 ","pages":"Article 102627"},"PeriodicalIF":1.2,"publicationDate":"2025-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143808775","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}

{"title":"Cyclically covering subspaces of Fqn","authors":"Meng Sun,&nbsp;Changli Ma,&nbsp;Liwei Zeng","doi":"10.1016/j.ffa.2025.102625","DOIUrl":"10.1016/j.ffa.2025.102625","url":null,"abstract":"<div><div>Let <em>q</em> be a prime power, <em>n</em> be a positive integer, and <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> be the <em>n</em>-dimensional row vector space over finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>. We say a subspace <em>U</em> of <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> is cyclically covering if the union of the cyclic shifts of <em>U</em> is equal to <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math></span>. Recently, the largest possible codimension of a cyclically covering subspace of <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math></span>, denoted by <span><math><msub><mrow><mi>h</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span>, has attracted the attention of many scholars. In this paper, we introduce cyclically covering subspaces of finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span>. By virtue of the theory of direct sum decomposition of finite fields, we describe a method for constructing cyclically covering subspaces of <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span>, and determine the value of <span><math><msub><mrow><mi>h</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> for some special <em>n</em>. In particular, we prove <span><math><msub><mrow><mi>h</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mn>21</mn><mo>)</mo><mo>=</mo><mn>4</mn></math></span>. Finally, several lower bounds of <span><math><msub><mrow><mi>h</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> are given when <span><math><mo>(</mo><mi>q</mi><mo>,</mo><mi>n</mi><mo>)</mo><mo>=</mo><mn>1</mn></math></span>, which generalizes results of the existing results in <span><span>[2]</span></span>.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"106 ","pages":"Article 102625"},"PeriodicalIF":1.2,"publicationDate":"2025-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143808774","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}

{"title":"The study on three classes of permutation quadrinomials over finite fields of odd characteristic","authors":"Changhui Chen ,&nbsp;Haibin Kan ,&nbsp;Jie Peng ,&nbsp;Hengtai Wang ,&nbsp;Lijing Zheng","doi":"10.1016/j.ffa.2025.102626","DOIUrl":"10.1016/j.ffa.2025.102626","url":null,"abstract":"<div><div>Let <em>p</em> be an odd prime and <em>n</em> a positive integer. This paper focuses on the investigation of permutation quadrinomials with generalized Niho exponents over finite fields of odd characteristic. Inspired by the works in <span><span>[10]</span></span>, we study two classes of permutation quadrinomials over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span> and a class of permutation quadrinomials over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mn>5</mn></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span>. Finally, we verify that permutation quadrinomials presented in this paper are multiplicative inequivalent to known ones.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"105 ","pages":"Article 102626"},"PeriodicalIF":1.2,"publicationDate":"2025-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143808770","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}

{"title":"Some classes of permutation pentanomials","authors":"Zhiguo Ding ,&nbsp;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 <span><math><mi>p</mi><mo>≠</mo><mn>3</mn></math></span> and each power <span><math><mi>q</mi><mo>=</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>k</mi></mrow></msup></math></span>, we present two large classes of permutation polynomials over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub></math></span> of the form <span><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></span> which have at most five terms, where <span><math><mi>B</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> is a polynomial with coefficients in <span><math><mo>{</mo><mn>1</mn><mo>,</mo><mo>−</mo><mn>1</mn><mo>}</mo></math></span>. The special case <span><math><mi>p</mi><mo>=</mo><mn>2</mn></math></span> of our results comprises a vast generalization of 76 recent results and conjectures in the literature. In case <span><math><mi>p</mi><mo>&gt;</mo><mn>2</mn></math></span>, 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}

{"title":"On self-dual completely regular codes with covering radius ρ ≤ 3","authors":"J. Borges ,&nbsp;V. Zinoviev","doi":"10.1016/j.ffa.2025.102617","DOIUrl":"10.1016/j.ffa.2025.102617","url":null,"abstract":"<div><div>We give a complete classification of self-dual completely regular codes with covering radius <span><math><mi>ρ</mi><mo>≤</mo><mn>3</mn></math></span>. For <span><math><mi>ρ</mi><mo>=</mo><mn>1</mn></math></span> the results are almost trivial. For <span><math><mi>ρ</mi><mo>=</mo><mn>2</mn></math></span>, by using properties of the more general class of uniformly packed codes in the wide sense, we show that there are two sporadic such codes, of length 8, and an infinite family, of length 4, apart from the direct sum of two self-dual completely regular codes with <span><math><mi>ρ</mi><mo>=</mo><mn>1</mn></math></span>, each one. For <span><math><mi>ρ</mi><mo>=</mo><mn>3</mn></math></span>, in some cases, we use similar techniques to the ones used for <span><math><mi>ρ</mi><mo>=</mo><mn>2</mn></math></span>. However, for some other cases we use different methods, namely, the Pless power moments which allow to us to discard several possibilities. We show that there are only two self-dual completely regular codes with <span><math><mi>ρ</mi><mo>=</mo><mn>3</mn></math></span> and <span><math><mi>d</mi><mo>≥</mo><mn>3</mn></math></span>, which are both ternary: the extended ternary Golay code and the direct sum of three ternary Hamming codes of length 4. Therefore, any self-dual completely regular code with <span><math><mi>d</mi><mo>≥</mo><mn>3</mn></math></span> and <span><math><mi>ρ</mi><mo>=</mo><mn>3</mn></math></span> is ternary and has length 12.</div><div>We provide the intersection arrays for all such codes.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"105 ","pages":"Article 102617"},"PeriodicalIF":1.2,"publicationDate":"2025-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143704076","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}

{"title":"The differential uniformity of the power functions xpn+52 over Fpn","authors":"Wenping Yuan ,&nbsp;Xiaoni Du ,&nbsp;Huan Zhou ,&nbsp;Xingbin Qiao","doi":"10.1016/j.ffa.2025.102622","DOIUrl":"10.1016/j.ffa.2025.102622","url":null,"abstract":"<div><div>Cryptographic functions with low differential uniformity have important applications in designing S-box in the block ciphers. In this paper, we mainly investigate the differential uniformity <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mi>F</mi></mrow></msub></math></span> on a new class of power mappings <span><math><mi>F</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msup><mrow><mi>x</mi></mrow><mrow><mfrac><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>+</mo><mn>5</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup></math></span> over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span> with <em>p</em> being an odd prime and <em>n</em> being a positive integer. More precisely, for <span><math><mi>p</mi><mo>=</mo><mn>3</mn></math></span>, the differential uniformity and the differential spectrum of <em>F</em> have been determined explicitly. The results indicate that <em>F</em> is a locally-PN function with differentially <span><math><mfrac><mrow><msup><mrow><mn>3</mn></mrow><mrow><mi>n</mi></mrow></msup><mo>+</mo><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></math></span>-uniform when <em>n</em> is odd and a locally-APN function with differentially <span><math><mfrac><mrow><msup><mrow><mn>3</mn></mrow><mrow><mi>n</mi></mrow></msup><mo>+</mo><mn>3</mn></mrow><mrow><mn>4</mn></mrow></mfrac></math></span>-uniform when <em>n</em> is even. Then, for <span><math><mi>p</mi><mo>=</mo><mn>5</mn></math></span>, we prove that <em>F</em> is APN for even <em>n</em> and <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mi>F</mi></mrow></msub><mo>=</mo><mn>6</mn></math></span> for odd <em>n</em> through specific differential equations and quadratic character over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mn>5</mn></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span>. The method is different from the existing one. Moreover, for primes <span><math><mi>p</mi><mo>&gt;</mo><mn>5</mn></math></span>, we show that <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mi>F</mi></mrow></msub><mo>≤</mo><mn>5</mn></math></span> when <span><math><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>≡</mo><mn>3</mn><mspace></mspace><mo>(</mo><mspace></mspace><mrow><mi>mod</mi></mrow><mspace></mspace><mspace></mspace><mn>4</mn><mo>)</mo></math></span> and <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mi>F</mi></mrow></msub><mo>≤</mo><mn>8</mn></math></span> when <span><math><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>≡</mo><mn>1</mn><mspace></mspace><mo>(</mo><mspace></mspace><mrow><mi>mod</mi></mrow><mspace></mspace><mspace></mspace><mn>4</mn><mo>)</mo></math></span>.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"105 ","pages":"Article 102622"},"PeriodicalIF":1.2,"publicationDate":"2025-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143716154","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}

{"title":"Zeros of a system of diagonal polynomials over finite fields","authors":"Yulu Feng","doi":"10.1016/j.ffa.2025.102623","DOIUrl":"10.1016/j.ffa.2025.102623","url":null,"abstract":"&lt;div&gt;&lt;div&gt;Let &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; be the finite field of characteristic &lt;em&gt;p&lt;/em&gt;, having &lt;span&gt;&lt;math&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; elements and let &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;/math&gt;&lt;/span&gt; be the unit group of &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;. Let &lt;span&gt;&lt;math&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; be the number of &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;-rational points of the affine algebraic variety defined by the simultaneous vanishing of the diagonal polynomials &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mo&gt;⋯&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, where &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; is a nonnegative integer for &lt;span&gt;&lt;math&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. By using properties of Teichmüller representations and the Stickelberger relation applied by Ax and Wan, we show that&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ord&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mo&gt;⌈&lt;/mo&gt;&lt;munderover&gt;&lt;mo&gt;∑&lt;/mo&gt;&lt;mrow&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/munderover&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;munder&gt;&lt;mi&gt;max&lt;/mi&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/munder&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;⌉&lt;/mo&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; if &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;max&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"106 ","pages":"Article 102623"},"PeriodicalIF":1.2,"publicationDate":"2025-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143704040","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}