Finite Fields and Their Applications最新文献

{"title":"On the index of power compositional polynomials","authors":"Sumandeep Kaur ,&nbsp;Surender Kumar ,&nbsp;László Remete","doi":"10.1016/j.ffa.2025.102642","DOIUrl":"10.1016/j.ffa.2025.102642","url":null,"abstract":"<div><div>The index of a monic irreducible polynomial <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>∈</mo><mi>Z</mi><mo>[</mo><mi>x</mi><mo>]</mo></math></span> having a root <em>θ</em> is the index <span><math><mo>[</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>K</mi></mrow></msub><mo>:</mo><mi>Z</mi><mo>[</mo><mi>θ</mi><mo>]</mo><mo>]</mo></math></span> where <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>K</mi></mrow></msub></math></span> is the ring of algebraic integers of the number field <span><math><mi>K</mi><mo>=</mo><mi>Q</mi><mo>(</mo><mi>θ</mi><mo>)</mo></math></span>. If <span><math><mo>[</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>K</mi></mrow></msub><mo>:</mo><mi>Z</mi><mo>[</mo><mi>θ</mi><mo>]</mo><mo>]</mo><mo>=</mo><mn>1</mn></math></span>, then <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> is monogenic. In this paper, we give necessary and sufficient conditions for a monic irreducible power compositional polynomial <span><math><mi>f</mi><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>)</mo></math></span> belonging to <span><math><mi>Z</mi><mo>[</mo><mi>x</mi><mo>]</mo></math></span>, to be monogenic. As an application of our results, for a polynomial <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>+</mo><mi>A</mi><mo>⋅</mo><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>∈</mo><mi>Z</mi><mo>[</mo><mi>x</mi><mo>]</mo></math></span>, with <span><math><mi>d</mi><mo>&gt;</mo><mn>1</mn><mo>,</mo><mi>deg</mi><mo>⁡</mo><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>&lt;</mo><mi>d</mi></math></span> and <span><math><mo>|</mo><mi>h</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>|</mo><mo>=</mo><mn>1</mn></math></span>, we prove that for each positive integer <em>k</em> with <span><math><mi>rad</mi><mo>(</mo><mi>k</mi><mo>)</mo><mo>|</mo><mi>rad</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span>, the power compositional polynomial <span><math><mi>f</mi><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>)</mo></math></span> is monogenic if and only if <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> is monogenic, provided that <span><math><mi>f</mi><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>)</mo></math></span> is irreducible. At the end of the paper, we give infinite families of polynomials as examples.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"107 ","pages":"Article 102642"},"PeriodicalIF":1.2,"publicationDate":"2025-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143918442","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 a theorem of Kara and Klyachko","authors":"Sanmin Wang","doi":"10.1016/j.ffa.2025.102645","DOIUrl":"10.1016/j.ffa.2025.102645","url":null,"abstract":"<div><div>There exists a finite set of pairs <span><math><mo>(</mo><msub><mrow><mi>Γ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>N</mi><mo>)</mo><msub><mrow><mi>τ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>Γ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>N</mi><mo>)</mo><msub><mrow><mi>τ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> of points of the modular curve <span><math><msub><mrow><mi>X</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>N</mi><mo>)</mo></math></span> with <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>N</mi><mo>)</mo><msub><mrow><mi>τ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≠</mo><msub><mrow><mi>Γ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>N</mi><mo>)</mo><msub><mrow><mi>τ</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> but <span><math><mo>(</mo><mi>j</mi><mo>(</mo><msub><mrow><mi>τ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>,</mo><mi>j</mi><mo>(</mo><mi>N</mi><msub><mrow><mi>τ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>)</mo><mo>=</mo><mo>(</mo><mi>j</mi><mo>(</mo><msub><mrow><mi>τ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>,</mo><mi>j</mi><mo>(</mo><mi>N</mi><msub><mrow><mi>τ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>)</mo></math></span>, which are the singularities of the plane model <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>N</mi><mo>)</mo></math></span> of <span><math><msub><mrow><mi>X</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>N</mi><mo>)</mo></math></span>. A theorem of Kara and Klyachko gives a characterization of these pairs of points of <span><math><msub><mrow><mi>X</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>N</mi><mo>)</mo></math></span>. However, their proof for this theorem contains an erroneous assertion. Following their idea, we give an elementary proof for this theorem in details.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"107 ","pages":"Article 102645"},"PeriodicalIF":1.2,"publicationDate":"2025-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143912196","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 second generalized covering radius of binary primitive double-error-correcting BCH codes","authors":"Lev Yohananov ,&nbsp;Moshe Schwartz","doi":"10.1016/j.ffa.2025.102648","DOIUrl":"10.1016/j.ffa.2025.102648","url":null,"abstract":"<div><div>We completely determine the second covering radius for binary primitive double-error-correcting BCH codes. As part of this process, we provide a lower bound on the second covering radius for binary primitive BCH codes correcting more than two errors.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"107 ","pages":"Article 102648"},"PeriodicalIF":1.2,"publicationDate":"2025-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143918440","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 the b-symbol weights of linear codes for large b","authors":"Dongmei Huang ,&nbsp;Qunying Liao ,&nbsp;Gaohua Tang ,&nbsp;Shixin Zhu","doi":"10.1016/j.ffa.2025.102647","DOIUrl":"10.1016/j.ffa.2025.102647","url":null,"abstract":"<div><div>Recently, <em>b</em>-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 <em>b</em>-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 <em>b</em>-symbol weight for <span><math><mi>b</mi><mo>≥</mo><mn>3</mn></math></span>. In this paper, we mainly consider the <em>b</em>-symbol weight of linear codes for large <em>b</em>. Firstly, we establish a direct correspondence between the <em>b</em>-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 <em>b</em>-symbol weight, which partially settles a conjecture of Shi, Zhu and Helleseth. Furthermore, we discover that the minimum <em>b</em>-symbol distance of a <span><math><mo>[</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>]</mo></math></span> cyclic code is equal to the minimum <em>b</em>-symbol distance of certain linear code of parameters <span><math><mo>[</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>−</mo><mi>b</mi><mo>]</mo></math></span> and prove that every <span><math><mo>[</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>]</mo></math></span> cyclic code is always <em>b</em>-symbol MDS for <span><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></span>, where <span><math><msup><mrow><mi>d</mi></mrow><mrow><mo>⊥</mo></mrow></msup></math></span> is the minimum distance of its dual. When <span><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></span> we also determine its <em>b</em>-symbol weight distribution. Finally, we investigate the minimum <em>b</em>-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}

{"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 <em>φ</em> be a form (homogeneous polynomial) of degree <em>d</em> in <span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span> variables over a field <em>F</em>. We call <em>φ</em> anisotropic over <em>F</em>, if the equality <span><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></span> with <span><math><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∈</mo><mi>F</mi></math></span> implies <span><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></span>. Otherwise <em>φ</em> is called isotropic. Assume that <span><math><mi>L</mi><mo>/</mo><mi>F</mi></math></span> be a finite field extension, the numbers <em>d</em> and <span><math><mo>[</mo><mi>L</mi><mo>:</mo><mi>F</mi><mo>]</mo></math></span> are coprime, and the form <em>φ</em> is anisotropic and diagonal. Does the form <em>φ</em> remain anisotropic over <em>L</em>? 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 <span><math><mi>F</mi><mo>=</mo><mi>Q</mi></math></span>. 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 <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> 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}

{"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}