Finite Fields and Their Applications最新文献

{"title":"Construction of Galois self-orthogonal MDS codes with larger dimensions","authors":"Ruhao Wan, Shixin Zhu","doi":"10.1016/j.ffa.2025.102665","DOIUrl":"10.1016/j.ffa.2025.102665","url":null,"abstract":"<div><div>Let <span><math><mi>q</mi><mo>=</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>m</mi></mrow></msup></math></span> be a prime power, <em>e</em> be an integer with <span><math><mn>0</mn><mo>≤</mo><mi>e</mi><mo>≤</mo><mi>m</mi><mo>−</mo><mn>1</mn></math></span> and <span><math><mi>s</mi><mo>=</mo><mi>gcd</mi><mo>⁡</mo><mo>(</mo><mi>e</mi><mo>,</mo><mi>m</mi><mo>)</mo></math></span>. In this paper, for a vector <span><math><mi>v</mi><mo>∈</mo><msup><mrow><mo>(</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup><mo>)</mo></mrow><mrow><mi>n</mi></mrow></msup></math></span> and a <em>q</em>-ary linear code <span><math><mi>C</mi></math></span>, we give some necessary and sufficient conditions for the equivalent code <span><math><msub><mrow><mi>Φ</mi></mrow><mrow><mi>v</mi></mrow></msub><mo>(</mo><mi>C</mi><mo>)</mo></math></span> of <span><math><mi>C</mi></math></span> and the extended code of <span><math><msub><mrow><mi>Φ</mi></mrow><mrow><mi>v</mi></mrow></msub><mo>(</mo><mi>C</mi><mo>)</mo></math></span> to be <em>e</em>-Galois self-orthogonal. We then directly obtain some necessary and sufficient conditions for (extended) generalized Reed-Solomon (GRS and EGRS) codes to be <em>e</em>-Galois self-orthogonal. From this we show that if <span><math><mi>k</mi><mo>≥</mo><mi>min</mi><mo>⁡</mo><mo>{</mo><mi>max</mi><mo>⁡</mo><mo>{</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>e</mi></mrow></msup><mo>,</mo><mo>⌈</mo><mfrac><mrow><mi>n</mi><mo>+</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>e</mi></mrow></msup></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>e</mi></mrow></msup><mo>+</mo><mn>1</mn></mrow></mfrac><mo>⌉</mo><mo>}</mo><mo>,</mo><mi>max</mi><mo>⁡</mo><mo>{</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>m</mi><mo>−</mo><mi>e</mi></mrow></msup><mo>,</mo><mo>⌈</mo><mfrac><mrow><mi>n</mi><mo>+</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>m</mi><mo>−</mo><mi>e</mi></mrow></msup></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>m</mi><mo>−</mo><mi>e</mi></mrow></msup><mo>+</mo><mn>1</mn></mrow></mfrac><mo>⌉</mo><mo>}</mo><mo>}</mo></math></span>, there is no <span><math><msub><mrow><mo>[</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>]</mo></mrow><mrow><mi>q</mi></mrow></msub></math></span> <em>e</em>-Galois self-orthogonal (extended) GRS code. Furthermore, for all possible <em>e</em> satisfying <span><math><mn>0</mn><mo>≤</mo><mi>e</mi><mo>≤</mo><mi>m</mi><mo>−</mo><mn>1</mn></math></span>, we classify them into three cases (1) <span><math><mfrac><mrow><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></math></span> odd and <em>p</em> even; (2) <span><math><mfrac><mrow><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></math></span> odd and <em>p</em> odd; (3) <span><math><mfrac><mrow><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></math></span> even, and construct several new classes of <em>e</em>-Galois self-orthogonal maximum distance separable (MDS) codes. It is worth noting that our <em>e</em>-Galois self-orthogonal MDS","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"108 ","pages":"Article 102665"},"PeriodicalIF":1.2,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144222654","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 compositional inverses of permutation polynomials of the form ∑i=1kbi(xpm+x+δ)si−x over Fp2m","authors":"Danyao Wu ,&nbsp;Pingzhi Yuan ,&nbsp;Huanhuan Guan ,&nbsp;Juan Li","doi":"10.1016/j.ffa.2025.102681","DOIUrl":"10.1016/j.ffa.2025.102681","url":null,"abstract":"<div><div>In this paper, we present the compositional inverses of several classes permutation polynomials of the form <span><math><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>k</mi></mrow></msubsup><msub><mrow><mi>b</mi></mrow><mrow><mi>i</mi></mrow></msub><msup><mrow><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>m</mi></mrow></msup></mrow></msup><mo>+</mo><mi>x</mi><mo>+</mo><mi>δ</mi><mo>)</mo></mrow><mrow><msub><mrow><mi>s</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msup><mo>−</mo><mi>x</mi></math></span> over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn><mi>m</mi></mrow></msup></mrow></msub></math></span>, where for <span><math><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>k</mi></math></span>, <span><math><msub><mrow><mi>s</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><mi>m</mi></math></span> are positive integers, <span><math><msub><mrow><mi>b</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><mi>δ</mi><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn><mi>m</mi></mrow></msup></mrow></msub></math></span>, and <em>p</em> is prime.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"108 ","pages":"Article 102681"},"PeriodicalIF":1.2,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144261668","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":"Permutation polynomials of finite fields of even characteristic from character sums","authors":"Ruikai Chen ,&nbsp;Sihem Mesnager","doi":"10.1016/j.ffa.2025.102684","DOIUrl":"10.1016/j.ffa.2025.102684","url":null,"abstract":"<div><div>In this paper, we investigate permutation polynomials over the 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> with <span><math><mi>q</mi><mo>=</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup></math></span>, focusing on those in the form <span><math><mi>Tr</mi><mo>(</mo><mi>A</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>q</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>)</mo><mo>+</mo><mi>L</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span>, where <span><math><mi>A</mi><mo>∈</mo><msubsup><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow><mrow><mo>⁎</mo></mrow></msubsup></math></span> and <em>L</em> is a 2-linear polynomial over <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 calculating certain character sums, we characterize these permutation polynomials and provide additional constructions.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"108 ","pages":"Article 102684"},"PeriodicalIF":1.2,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144513839","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":"A tighter bound on the minimum distances for an infinite family of binary BCH codes and its generalization","authors":"Haodong Lu,&nbsp;Xuan Wang,&nbsp;Minjia Shi","doi":"10.1016/j.ffa.2025.102628","DOIUrl":"10.1016/j.ffa.2025.102628","url":null,"abstract":"<div><div>In this paper, we improve the bound on the minimum distance for the family of binary cyclic codes proposed by Sun et al. (2024) <span><span>[8]</span></span>. The 3-ary analogue is also studied in this paper, which is a nice family of ternary cyclic codes that contains some best known linear codes, and this family has a better lower bound on minimum distance than that of codes proposed by Chen et al. (2023) <span><span>[2]</span></span>.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"108 ","pages":"Article 102628"},"PeriodicalIF":1.2,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143923884","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":"Counting irreducible polynomials with restricted coefficients","authors":"Kaimin Cheng","doi":"10.1016/j.ffa.2025.102691","DOIUrl":"10.1016/j.ffa.2025.102691","url":null,"abstract":"<div><div>Let <em>q</em> be a prime power, and let <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> denote the finite field with <em>q</em> elements. Consider a positive integer <em>n</em>, and let <span><math><mi>R</mi><mo>=</mo><msubsup><mrow><mo>{</mo><msub><mrow><mi>R</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>}</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msubsup></math></span> be a family of subsets of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>. Define <span><math><mi>N</mi><mo>(</mo><mi>R</mi><mo>,</mo><mi>n</mi><mo>)</mo></math></span> as the number of monic irreducible polynomials of degree <em>n</em> over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> where the coefficient of each non-leading term <span><math><msup><mrow><mi>T</mi></mrow><mrow><mi>i</mi></mrow></msup></math></span> lies in <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>∖</mo><msub><mrow><mi>R</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>. In this paper, we provide an asymptotic formula for <span><math><mi>N</mi><mo>(</mo><mi>R</mi><mo>,</mo><mi>n</mi><mo>)</mo></math></span>, extending a result of Porritt to a more general case.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"108 ","pages":"Article 102691"},"PeriodicalIF":1.2,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144570385","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":"Partial difference sets from unions of cyclotomic classes","authors":"Ka Hin Leung ,&nbsp;Koji Momihara ,&nbsp;Qing Xiang","doi":"10.1016/j.ffa.2025.102661","DOIUrl":"10.1016/j.ffa.2025.102661","url":null,"abstract":"<div><div>In their study of two-weight irreducible cyclic codes, Schmidt and White (2002) obtained a necessary and sufficient condition on <span><math><mo>(</mo><mi>q</mi><mo>,</mo><mi>N</mi><mo>)</mo></math></span> under which the multiplicative subgroup of index <em>N</em> of the finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> forms a regular partial difference set (PDS) in the additive group of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>. They also found 11 sporadic examples by a computer search aside from two known infinite families of PDS. In this paper, we study the problem of determining for which <span><math><mo>(</mo><mi>q</mi><mo>,</mo><mi>N</mi><mo>)</mo></math></span> a union of multiple cosets of the multiplicative subgroup of index <em>N</em> of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> forms a regular PDS in the additive group of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>. Building on the work of Schmidt and White, we find a necessary and sufficient numerical condition on the parameters <span><math><mo>(</mo><mi>q</mi><mo>,</mo><mi>N</mi><mo>)</mo></math></span> for unions of multiple cyclotomic classes to form regular PDS in <span><math><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>,</mo><mo>+</mo><mo>)</mo></math></span>. We then apply the theorem to the situation where unions of a small number of classes are selected in a structured manner. We obtain a new infinite family of regular PDS not belonging to previously known families, and two sporadic examples of regular PDS (one of which is new) with the help of a computer research. We further propose a conjecture analogous to the Schmidt-White conjecture proposed in their 2002 paper.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"108 ","pages":"Article 102661"},"PeriodicalIF":1.2,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144190116","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":"Expansion properties of polynomials over finite fields","authors":"Nuno Arala ,&nbsp;Sam Chow","doi":"10.1016/j.ffa.2025.102687","DOIUrl":"10.1016/j.ffa.2025.102687","url":null,"abstract":"<div><div>We establish expansion properties for suitably generic polynomials of degree <em>d</em> in <span><math><mi>d</mi><mo>+</mo><mn>1</mn></math></span> variables over finite fields. In particular, we show that if <span><math><mi>P</mi><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><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>d</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>]</mo></math></span> is a polynomial of degree <em>d</em>, whose coefficients avoid the zero locus of some explicit polynomial of degree <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mn>1</mn><mo>)</mo></math></span>, and <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>d</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>⊆</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> are suitably large, then <span><math><mo>|</mo><mi>P</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>d</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>)</mo><mo>|</mo><mo>=</mo><mi>q</mi><mo>−</mo><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></math></span>. Our methods rely on a higher-degree extension of a result of Vinh on point–line incidences over a finite field.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"108 ","pages":"Article 102687"},"PeriodicalIF":1.2,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144570241","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":"Deep holes of twisted Reed-Solomon codes","authors":"Weijun Fang ,&nbsp;Jingke Xu ,&nbsp;Ruiqi Zhu","doi":"10.1016/j.ffa.2025.102680","DOIUrl":"10.1016/j.ffa.2025.102680","url":null,"abstract":"<div><div>The deep holes of a linear code are the vectors that achieve the maximum error distance (covering radius) to the code. Determining the covering radius and deep holes of linear codes is a fundamental problem in coding theory. In this paper, we investigate the problem of deep holes of twisted Reed-Solomon codes. The covering radius and a standard class of deep holes of twisted Reed-Solomon codes <span><math><msub><mrow><mi>TRS</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>A</mi><mo>,</mo><mi>θ</mi><mo>)</mo></math></span> are obtained for a general evaluation set <span><math><mi>A</mi><mo>⊆</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>. Furthermore, we consider the problem of determining all deep holes of the full-length twisted Reed-Solomon codes <span><math><msub><mrow><mi>TRS</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>,</mo><mi>θ</mi><mo>)</mo></math></span>. For even <em>q</em>, by utilizing the polynomial method and Gauss sums over finite fields, we prove that the standard deep holes are all the deep holes of <span><math><msub><mrow><mi>TRS</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>,</mo><mi>θ</mi><mo>)</mo></math></span> with <span><math><mfrac><mrow><mn>3</mn><mi>q</mi><mo>−</mo><mn>4</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>q</mi><mo>−</mo><mn>4</mn></math></span>. For odd <em>q</em>, we adopt a different method and employ the results on some equations over finite fields to show that there are also no other deep holes of <span><math><msub><mrow><mi>TRS</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>,</mo><mi>θ</mi><mo>)</mo></math></span> with <span><math><mfrac><mrow><mn>3</mn><mi>q</mi><mo>+</mo><mn>3</mn><msqrt><mrow><mi>q</mi></mrow></msqrt><mo>−</mo><mn>7</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>q</mi><mo>−</mo><mn>4</mn></math></span>. In addition, for the boundary cases of <span><math><mi>k</mi><mo>=</mo><mi>q</mi><mo>−</mo><mn>3</mn><mo>,</mo><mi>q</mi><mo>−</mo><mn>2</mn></math></span> and <span><math><mi>q</mi><mo>−</mo><mn>1</mn></math></span>, we completely determine their deep holes using results on certain character sums.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"108 ","pages":"Article 102680"},"PeriodicalIF":1.2,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144290832","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":"Projective symplectic groups of genus one and two","authors":"Rezhna M. Hussein,&nbsp;Haval M. Mohammed Salih","doi":"10.1016/j.ffa.2025.102689","DOIUrl":"10.1016/j.ffa.2025.102689","url":null,"abstract":"<div><div>In this article, we provide a comprehensive classification of all primitive genus one and genus two systems of the finite group <em>G</em> with <span><math><mi>P</mi><mi>S</mi><mi>p</mi><mo>(</mo><mn>4</mn><mo>,</mo><mi>q</mi><mo>)</mo><mo>≤</mo><mi>G</mi><mo>≤</mo><mi>A</mi><mi>u</mi><mi>t</mi><mo>(</mo><mi>P</mi><mi>S</mi><mi>p</mi><mo>(</mo><mn>4</mn><mo>,</mo><mi>q</mi><mo>)</mo><mo>)</mo></math></span>, where <em>q</em> is a prime power. Also, we use computational tools to show that <em>G</em> possesses no genus <em>g</em> group if <span><math><mi>q</mi><mo>&gt;</mo><mn>5</mn></math></span> where <span><math><mi>g</mi><mo>=</mo><mn>1</mn></math></span>, and 2.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"108 ","pages":"Article 102689"},"PeriodicalIF":1.2,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144569057","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":"Decoding up to Hartmann–Tzeng and Roos bounds for rank codes","authors":"José Manuel Muñoz","doi":"10.1016/j.ffa.2025.102676","DOIUrl":"10.1016/j.ffa.2025.102676","url":null,"abstract":"<div><div>A class of linear block codes which simultaneously generalizes Gabidulin codes and a class of skew cyclic codes is defined. For these codes, both a Hartmann–Tzeng-like bound and a Roos-like bound, with respect to their rank distance, are described, and corresponding nearest-neighbor decoding algorithms are presented. Additional necessary conditions so that decoding can be done up to the described bounds are studied. Subfield subcodes and interleaved codes from the considered class of codes are also described, since they allow an unbounded length for the codes, providing a decoding algorithm for them; additionally, both approaches are shown to yield equivalent codes with respect to the rank metric.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"108 ","pages":"Article 102676"},"PeriodicalIF":1.2,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144253583","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}