Pub Date : 2026-06-01Epub Date: 2026-02-09DOI: 10.1016/j.ffa.2026.102811
Ruihua Shen , Xianping Liu , Xiaofang Xu
Sparse permutation polynomials over finite fields have attracted more and more researchers' attention due to their simple algebraic structure and wide applications in many areas. In this paper, we propose three classes of permutation pentanomials over the finite field with even characteristic and provide the explicit expression of the compositional inverse for one of them. Furthermore, two classes of permutation hexanomials over are constructed, and their explicit expressions of the compositional inverses are determined. The results are derived by investigating the solutions of some equations, employing the resultant elimination and multivariate methods.
{"title":"Some new classes of permutation pentanomials and hexanomials over Fq3 with even characteristic","authors":"Ruihua Shen , Xianping Liu , Xiaofang Xu","doi":"10.1016/j.ffa.2026.102811","DOIUrl":"10.1016/j.ffa.2026.102811","url":null,"abstract":"<div><div>Sparse permutation polynomials over finite fields have attracted more and more researchers' attention due to their simple algebraic structure and wide applications in many areas. In this paper, we propose three classes of permutation pentanomials over the finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>3</mn></mrow></msup></mrow></msub></math></span> with even characteristic and provide the explicit expression of the compositional inverse for one of them. Furthermore, two classes of permutation hexanomials over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>3</mn></mrow></msup></mrow></msub></math></span> are constructed, and their explicit expressions of the compositional inverses are determined. The results are derived by investigating the solutions of some equations, employing the resultant elimination and multivariate methods.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"112 ","pages":"Article 102811"},"PeriodicalIF":1.2,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146189252","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}
Pub Date : 2026-06-01Epub Date: 2026-01-19DOI: 10.1016/j.ffa.2026.102793
Owen J. Brison , J. Eurico Nogueira
We study the behaviour with respect to the operation of lifting of the unit g-sequence, where is a primitive polynomial. We also study the unit f-sequence for primitive-based and show it is closely related to the lifted unit g-sequence where is based on .
{"title":"The unit f-sequence for primitive-based f","authors":"Owen J. Brison , J. Eurico Nogueira","doi":"10.1016/j.ffa.2026.102793","DOIUrl":"10.1016/j.ffa.2026.102793","url":null,"abstract":"<div><div>We study the behaviour with respect to the operation of lifting of the unit <em>g</em>-sequence, where <span><math><mi>g</mi><mo>(</mo><mi>t</mi><mo>)</mo></math></span> is a primitive polynomial. We also study the unit <em>f</em>-sequence for primitive-based <span><math><mi>f</mi><mo>(</mo><mi>t</mi><mo>)</mo></math></span> and show it is closely related to the lifted unit <em>g</em>-sequence where <span><math><mi>f</mi><mo>(</mo><mi>t</mi><mo>)</mo></math></span> is based on <span><math><mi>g</mi><mo>(</mo><mi>t</mi><mo>)</mo></math></span>.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"112 ","pages":"Article 102793"},"PeriodicalIF":1.2,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146039256","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}
Pub Date : 2026-06-01Epub Date: 2026-01-13DOI: 10.1016/j.ffa.2025.102789
Lavanya G, Anuradha Sharma
<div><div>Let <em>q</em> be a prime power, and let <em>m</em>, <em>v</em>, <em>t</em> be integers satisfying <span><math><mn>2</mn><mo>≤</mo><mi>t</mi><mo><</mo><mi>v</mi><mo>≤</mo><mi>m</mi></math></span> and <span><math><msup><mrow><mi>q</mi></mrow><mrow><mi>m</mi><mo>−</mo><mi>t</mi></mrow></msup><mo>></mo><mrow><mo>(</mo><mtable><mtr><mtd><mi>v</mi></mtd></mtr><mtr><mtd><mi>t</mi></mtd></mtr></mtable><mo>)</mo></mrow><mo>≥</mo><mn>3</mn></math></span>, where <span><math><mo>(</mo><mtable><mtr><mtd><mo>⋅</mo></mtd></mtr><mtr><mtd><mo>⋅</mo></mtd></mtr></mtable><mo>)</mo></math></span> denotes the binomial coefficient. Let <em>X</em> be a subset of <span><math><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>m</mi><mo>}</mo></math></span> with <span><math><mo>|</mo><mi>X</mi><mo>|</mo><mo>=</mo><mi>v</mi></math></span>. In this paper, we consider the set <span><math><mi>Δ</mi><mo>=</mo><mo>{</mo><mi>u</mi><mo>∈</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>m</mi></mrow></msubsup><mo>:</mo><mtext>supp</mtext><mo>(</mo><mi>u</mi><mo>)</mo><mo>⊆</mo><mi>X</mi><mtext> and </mtext><mi>w</mi><mo>(</mo><mi>u</mi><mo>)</mo><mo>≤</mo><mi>t</mi><mo>}</mo></math></span>, where <span><math><mtext>supp</mtext><mo>(</mo><mo>⋅</mo><mo>)</mo></math></span> denotes the support of a vector and <span><math><mi>w</mi><mo>(</mo><mo>⋅</mo><mo>)</mo></math></span> denotes the Hamming weight function. We first observe that the set Δ is a simplicial complex of <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>m</mi></mrow></msubsup></math></span> with support <span><math><mi>A</mi><mo>=</mo><mo>{</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>b</mi></mrow></msub><mo>}</mo></math></span> consisting of all distinct subsets of <em>X</em> with cardinality <em>t</em>. Note that <span><math><mi>b</mi><mo>=</mo><mrow><mo>(</mo><mtable><mtr><mtd><mi>v</mi></mtd></mtr><mtr><mtd><mi>t</mi></mtd></mtr></mtable><mo>)</mo></mrow><mo>≥</mo><mn>3</mn></math></span>, <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∖</mo><mo>(</mo><munder><mo>⋃</mo><mrow><mn>1</mn><mo>≤</mo><mi>j</mi><mo>(</mo><mo>≠</mo><mi>i</mi><mo>)</mo><mo>≤</mo><mi>b</mi></mrow></munder><msub><mrow><mi>A</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>)</mo><mo>=</mo><mo>∅</mo></math></span> for <span><math><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>b</mi></math></span>, and the pair <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo>)</mo></math></span> forms a trivial Steiner system. In this paper, we study linear codes over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> with defining sets <span><math><msup><mrow><mi>Δ</mi></mrow><mrow><mi>c</mi></mrow></msup><mo>=</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</
{"title":"Construction of several infinite families of linear codes with new parameters: Hamming weight enumerators and hull dimensions","authors":"Lavanya G, Anuradha Sharma","doi":"10.1016/j.ffa.2025.102789","DOIUrl":"10.1016/j.ffa.2025.102789","url":null,"abstract":"<div><div>Let <em>q</em> be a prime power, and let <em>m</em>, <em>v</em>, <em>t</em> be integers satisfying <span><math><mn>2</mn><mo>≤</mo><mi>t</mi><mo><</mo><mi>v</mi><mo>≤</mo><mi>m</mi></math></span> and <span><math><msup><mrow><mi>q</mi></mrow><mrow><mi>m</mi><mo>−</mo><mi>t</mi></mrow></msup><mo>></mo><mrow><mo>(</mo><mtable><mtr><mtd><mi>v</mi></mtd></mtr><mtr><mtd><mi>t</mi></mtd></mtr></mtable><mo>)</mo></mrow><mo>≥</mo><mn>3</mn></math></span>, where <span><math><mo>(</mo><mtable><mtr><mtd><mo>⋅</mo></mtd></mtr><mtr><mtd><mo>⋅</mo></mtd></mtr></mtable><mo>)</mo></math></span> denotes the binomial coefficient. Let <em>X</em> be a subset of <span><math><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>m</mi><mo>}</mo></math></span> with <span><math><mo>|</mo><mi>X</mi><mo>|</mo><mo>=</mo><mi>v</mi></math></span>. In this paper, we consider the set <span><math><mi>Δ</mi><mo>=</mo><mo>{</mo><mi>u</mi><mo>∈</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>m</mi></mrow></msubsup><mo>:</mo><mtext>supp</mtext><mo>(</mo><mi>u</mi><mo>)</mo><mo>⊆</mo><mi>X</mi><mtext> and </mtext><mi>w</mi><mo>(</mo><mi>u</mi><mo>)</mo><mo>≤</mo><mi>t</mi><mo>}</mo></math></span>, where <span><math><mtext>supp</mtext><mo>(</mo><mo>⋅</mo><mo>)</mo></math></span> denotes the support of a vector and <span><math><mi>w</mi><mo>(</mo><mo>⋅</mo><mo>)</mo></math></span> denotes the Hamming weight function. We first observe that the set Δ is a simplicial complex of <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>m</mi></mrow></msubsup></math></span> with support <span><math><mi>A</mi><mo>=</mo><mo>{</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>b</mi></mrow></msub><mo>}</mo></math></span> consisting of all distinct subsets of <em>X</em> with cardinality <em>t</em>. Note that <span><math><mi>b</mi><mo>=</mo><mrow><mo>(</mo><mtable><mtr><mtd><mi>v</mi></mtd></mtr><mtr><mtd><mi>t</mi></mtd></mtr></mtable><mo>)</mo></mrow><mo>≥</mo><mn>3</mn></math></span>, <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∖</mo><mo>(</mo><munder><mo>⋃</mo><mrow><mn>1</mn><mo>≤</mo><mi>j</mi><mo>(</mo><mo>≠</mo><mi>i</mi><mo>)</mo><mo>≤</mo><mi>b</mi></mrow></munder><msub><mrow><mi>A</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>)</mo><mo>=</mo><mo>∅</mo></math></span> for <span><math><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>b</mi></math></span>, and the pair <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo>)</mo></math></span> forms a trivial Steiner system. In this paper, we study linear codes over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> with defining sets <span><math><msup><mrow><mi>Δ</mi></mrow><mrow><mi>c</mi></mrow></msup><mo>=</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"112 ","pages":"Article 102789"},"PeriodicalIF":1.2,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145981219","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}
Pub Date : 2026-06-01Epub Date: 2026-02-03DOI: 10.1016/j.ffa.2026.102808
Alonso S. Castellanos , Erik Mendoza , Guilherme Tizziotti
In this work, we investigate generalized Weierstrass semigroups in arbitrary Kummer extensions of the rational function field . We analyze their structure and properties, with a particular emphasis on their maximal elements. Explicit descriptions of the sets of absolute and relative maximal elements within these semigroups are provided. Additionally, we apply our results to function fields of the maximal curves and , which cannot be covered by the Hermitian curve, and the Beelen-Montanucci curve. Our results generalize and unify several earlier contributions in the theory of Weierstrass semigroups, providing new perspectives on the relationship between these semigroups and function fields.
{"title":"On generalized Weierstrass semigroups in arbitrary Kummer extensions of Fq(x)","authors":"Alonso S. Castellanos , Erik Mendoza , Guilherme Tizziotti","doi":"10.1016/j.ffa.2026.102808","DOIUrl":"10.1016/j.ffa.2026.102808","url":null,"abstract":"<div><div>In this work, we investigate generalized Weierstrass semigroups in arbitrary Kummer extensions of the rational function field <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math></span>. We analyze their structure and properties, with a particular emphasis on their maximal elements. Explicit descriptions of the sets of absolute and relative maximal elements within these semigroups are provided. Additionally, we apply our results to function fields of the maximal curves <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>n</mi><mo>,</mo><mi>s</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>Y</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>s</mi></mrow></msub></math></span>, which cannot be covered by the Hermitian curve, and the Beelen-Montanucci curve. Our results generalize and unify several earlier contributions in the theory of Weierstrass semigroups, providing new perspectives on the relationship between these semigroups and function fields.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"112 ","pages":"Article 102808"},"PeriodicalIF":1.2,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146189255","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}
Pub Date : 2026-06-01Epub Date: 2026-02-03DOI: 10.1016/j.ffa.2026.102810
Satoru Fukasawa
This paper describes the arrangement of all Galois lines for a space model of the quotient of the Hermitian curve by an involution in odd characteristic, in terms of the geometry over finite fields. This paper also determines all Galois points for three plane models of this curve admitting three or more Galois points.
{"title":"Galois lines for a space model of the quotient of the Hermitian curve by an involution in odd characteristic","authors":"Satoru Fukasawa","doi":"10.1016/j.ffa.2026.102810","DOIUrl":"10.1016/j.ffa.2026.102810","url":null,"abstract":"<div><div>This paper describes the arrangement of all Galois lines for a space model of the quotient of the Hermitian curve by an involution in odd characteristic, in terms of the geometry over finite fields. This paper also determines all Galois points for three plane models of this curve admitting three or more Galois points.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"112 ","pages":"Article 102810"},"PeriodicalIF":1.2,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146189253","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}
Pub Date : 2026-06-01Epub Date: 2026-02-06DOI: 10.1016/j.ffa.2026.102813
Zhaole Li, Deng Tang
<div><div>Vectorial Boolean functions from <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> to <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msubsup></math></span> are fundamental objects in theoretical computer science and mathematics. Understanding their structure, particularly how well they can be approximated by low-degree functions, is crucial in various applications, including pseudorandomness, property testing, and cryptography. The Gowers uniformity norm, introduced in additive combinatorics, provides a powerful measure for these purposes, serving as a key indicator of the approximation and has significant applications in mathematics and theoretical computer science. While the Gowers <span><math><msub><mrow><mi>U</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> norm is well-understood, the analysis of higher-order structures, particularly the Gowers <span><math><msub><mrow><mi>U</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> norm for <span><math><mi>k</mi><mo>≥</mo><mn>3</mn></math></span>, poses significant challenges. Indeed, the computation of the Gowers <span><math><msub><mrow><mi>U</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> norm is intrinsically linked to the second-order differential spectrum of a function. However, determining this spectrum is a notoriously difficult problem, and to date, it has only been solved for a few specific cases, such as the inverse function over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span> for <span><math><mi>n</mi><mo>=</mo><mn>6</mn><mo>,</mo><mn>8</mn></math></span> and the APN permutation over <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>6</mn></mrow></msubsup></math></span>. In this paper, we investigate the Gowers <span><math><msub><mrow><mi>U</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> norms of vectorial Boolean functions, with specific focus on five classes of cubic highly nonlinear power permutations over finite fields. We provide a comprehensive analysis of the Gowers <span><math><msub><mrow><mi>U</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> norm for the Welch function, the Modified Welch function, the cubic Kasami function, the Bracken-Leander function, and the Cusick-Dobbertin function. By characterizing the distribution of solutions for all second-order derivatives of these functions, we derive exact expressions for their Gowers <span><math><msub><mrow><mi>U</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> norms. Our results reveal quantitative differences in higher-order uniformity among these functions, with the Modified Welch function exhibiting a smaller Gowers <span><math><msub><mrow><mi>U</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> norm compared to the Welch and cubic Kasami functions. An
{"title":"The Gowers U3 norm of five classes of power permutations","authors":"Zhaole Li, Deng Tang","doi":"10.1016/j.ffa.2026.102813","DOIUrl":"10.1016/j.ffa.2026.102813","url":null,"abstract":"<div><div>Vectorial Boolean functions from <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> to <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msubsup></math></span> are fundamental objects in theoretical computer science and mathematics. Understanding their structure, particularly how well they can be approximated by low-degree functions, is crucial in various applications, including pseudorandomness, property testing, and cryptography. The Gowers uniformity norm, introduced in additive combinatorics, provides a powerful measure for these purposes, serving as a key indicator of the approximation and has significant applications in mathematics and theoretical computer science. While the Gowers <span><math><msub><mrow><mi>U</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> norm is well-understood, the analysis of higher-order structures, particularly the Gowers <span><math><msub><mrow><mi>U</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> norm for <span><math><mi>k</mi><mo>≥</mo><mn>3</mn></math></span>, poses significant challenges. Indeed, the computation of the Gowers <span><math><msub><mrow><mi>U</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> norm is intrinsically linked to the second-order differential spectrum of a function. However, determining this spectrum is a notoriously difficult problem, and to date, it has only been solved for a few specific cases, such as the inverse function over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span> for <span><math><mi>n</mi><mo>=</mo><mn>6</mn><mo>,</mo><mn>8</mn></math></span> and the APN permutation over <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>6</mn></mrow></msubsup></math></span>. In this paper, we investigate the Gowers <span><math><msub><mrow><mi>U</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> norms of vectorial Boolean functions, with specific focus on five classes of cubic highly nonlinear power permutations over finite fields. We provide a comprehensive analysis of the Gowers <span><math><msub><mrow><mi>U</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> norm for the Welch function, the Modified Welch function, the cubic Kasami function, the Bracken-Leander function, and the Cusick-Dobbertin function. By characterizing the distribution of solutions for all second-order derivatives of these functions, we derive exact expressions for their Gowers <span><math><msub><mrow><mi>U</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> norms. Our results reveal quantitative differences in higher-order uniformity among these functions, with the Modified Welch function exhibiting a smaller Gowers <span><math><msub><mrow><mi>U</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> norm compared to the Welch and cubic Kasami functions. An","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"112 ","pages":"Article 102813"},"PeriodicalIF":1.2,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146189254","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}
Pub Date : 2026-06-01Epub Date: 2026-01-19DOI: 10.1016/j.ffa.2026.102798
Heming Cui , Xubo Zhao , Qiang Wang , Xiaoping Li , Tongjiang Yan
Linear codes with few weights have attracted significant interest due to their wide-ranging applications in secret sharing, authentication codes, association schemes, strongly regular graph, and some other fields. This paper focuses on unifying several existing construction methods for few-weight linear codes, extending the works of Wang et al. (2015) [24], Wu et al. (2019) [25], and Fang et al. (2023) [10]. In our code construction, we introduce a novel index set , whose cardinality and structural properties are shown to critically influence both the length and weight distribution of the resulting few-weight linear codes. By employing cyclotomic mappings and choosing the more general defining sets, several new classes of binary linear codes with at most three weights are constructed. Our framework subsumes all aforementioned constructions as special cases and enlarges the spectrum of attainable parameters. The weight distributions of the corresponding linear codes are also explicitly determined. We also demonstrate that some of the linear codes constructed in this paper are optimal in the sense that they have the best known parameters in the tables maintained by Markus Grassl and/or optimal in the sense that they meet certain bounds on linear codes.
{"title":"Binary linear codes with at most three weights from cyclotomic mappings","authors":"Heming Cui , Xubo Zhao , Qiang Wang , Xiaoping Li , Tongjiang Yan","doi":"10.1016/j.ffa.2026.102798","DOIUrl":"10.1016/j.ffa.2026.102798","url":null,"abstract":"<div><div>Linear codes with few weights have attracted significant interest due to their wide-ranging applications in secret sharing, authentication codes, association schemes, strongly regular graph, and some other fields. This paper focuses on unifying several existing construction methods for few-weight linear codes, extending the works of Wang et al. (2015) <span><span>[24]</span></span>, Wu et al. (2019) <span><span>[25]</span></span>, and Fang et al. (2023) <span><span>[10]</span></span>. In our code construction, we introduce a novel index set <span><math><mi>J</mi></math></span>, whose cardinality and structural properties are shown to critically influence both the length and weight distribution of the resulting few-weight linear codes. By employing cyclotomic mappings and choosing the more general defining sets, several new classes of binary linear codes with at most three weights are constructed. Our framework subsumes all aforementioned constructions as special cases and enlarges the spectrum of attainable parameters. The weight distributions of the corresponding linear codes are also explicitly determined. We also demonstrate that some of the linear codes constructed in this paper are optimal in the sense that they have the best known parameters in the tables maintained by Markus Grassl and/or optimal in the sense that they meet certain bounds on linear codes.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"112 ","pages":"Article 102798"},"PeriodicalIF":1.2,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146039255","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}
Pub Date : 2026-03-01Epub Date: 2025-12-01DOI: 10.1016/j.ffa.2025.102760
Liqin Qian , Minjia Shi , Xiwang Cao
In this paper, we propose several classes of permutation polynomials of the form over , where , and . The permutation behavior of the proposed polynomials is investigated by the AGW criterion and determination of the number of solutions to certain equations over . Based on an effective method proposed by Mesnager (2014), we construct several classes of involutions and further obtain some self-dual bent functions by employing three permutations of satisfying an algebraic property . Finally, it is worth pointing out that there exist examples of bent functions we obtained which do not belong to .
{"title":"Permutation polynomials and involutions over the finite field F22m","authors":"Liqin Qian , Minjia Shi , Xiwang Cao","doi":"10.1016/j.ffa.2025.102760","DOIUrl":"10.1016/j.ffa.2025.102760","url":null,"abstract":"<div><div>In this paper, we propose several classes of permutation polynomials of the form <span><math><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>t</mi></mrow></munderover><msup><mrow><mo>(</mo><msubsup><mrow><mi>Tr</mi></mrow><mrow><mi>m</mi></mrow><mrow><mn>2</mn><mi>m</mi></mrow></msubsup><msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></msup><mo>+</mo><msub><mrow><mi>δ</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>)</mo></mrow><mrow><msub><mrow><mi>s</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></msup><mo>+</mo><mi>L</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mn>2</mn><mi>m</mi></mrow></msup></mrow></msub></math></span>, where <span><math><mi>L</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>a</mi><msubsup><mrow><mi>Tr</mi></mrow><mrow><mi>m</mi></mrow><mrow><mn>2</mn><mi>m</mi></mrow></msubsup><mo>(</mo><mi>x</mi><mo>)</mo><mo>+</mo><mi>b</mi><mi>x</mi></math></span>, <span><math><mi>a</mi><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mn>2</mn><mi>m</mi></mrow></msup></mrow></msub></math></span> and <span><math><mi>b</mi><mo>∈</mo><msubsup><mrow><mi>F</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup></mrow><mrow><mo>⁎</mo></mrow></msubsup></math></span>. The permutation behavior of the proposed polynomials is investigated by the AGW criterion and determination of the number of solutions to certain equations over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mn>2</mn><mi>m</mi></mrow></msup></mrow></msub></math></span>. Based on an effective method proposed by Mesnager (2014), we construct several classes of involutions and further obtain some self-dual bent functions by employing three permutations of <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mn>2</mn><mi>m</mi></mrow></msup></mrow></msub></math></span> satisfying an algebraic property <span><math><mo>(</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>2</mn><mi>m</mi></mrow></msub><mo>)</mo></math></span>. Finally, it is worth pointing out that there exist examples of bent functions we obtained which do not belong to <span><math><mi>M</mi><msup><mrow><mi>M</mi></mrow><mrow><mi>#</mi></mrow></msup></math></span>.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"111 ","pages":"Article 102760"},"PeriodicalIF":1.2,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145684472","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}
Pub Date : 2026-03-01Epub Date: 2025-12-02DOI: 10.1016/j.ffa.2025.102766
I. Cardinali , L. Giuzzi
Let V be a vector space over the finite field with q elements and Λ be the image of the Segre geometry in under the Segre map. Consider the subvariety of Λ represented by the pure tensors with and such that . Regarding as a projective system of , we study the linear code arising from it. We show that is a minimal code and we determine its basic parameters, its full weight list and its linear automorphism group. We also give a geometrical characterization of its minimum and second lowest weight codewords as well as of some of the words of maximum weight.
{"title":"Linear codes arising from the point-hyperplane geometry-Part I: The Segre embedding","authors":"I. Cardinali , L. Giuzzi","doi":"10.1016/j.ffa.2025.102766","DOIUrl":"10.1016/j.ffa.2025.102766","url":null,"abstract":"<div><div>Let <em>V</em> be a vector space over the finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> with <em>q</em> elements and Λ be the image of the Segre geometry <span><math><mrow><mi>PG</mi></mrow><mo>(</mo><mi>V</mi><mo>)</mo><mo>⊗</mo><mrow><mi>PG</mi></mrow><mo>(</mo><msup><mrow><mi>V</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>)</mo></math></span> in <span><math><mrow><mi>PG</mi></mrow><mo>(</mo><mi>V</mi><mo>⊗</mo><msup><mrow><mi>V</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>)</mo></math></span> under the Segre map. Consider the subvariety <span><math><msub><mrow><mi>Λ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> of Λ represented by the pure tensors <span><math><mi>x</mi><mo>⊗</mo><mi>ξ</mi></math></span> with <span><math><mi>x</mi><mo>∈</mo><mi>V</mi></math></span> and <span><math><mi>ξ</mi><mo>∈</mo><msup><mrow><mi>V</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> such that <span><math><mi>ξ</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mn>0</mn></math></span>. Regarding <span><math><msub><mrow><mi>Λ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> as a projective system of <span><math><mrow><mi>PG</mi></mrow><mo>(</mo><mi>V</mi><mo>⊗</mo><msup><mrow><mi>V</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>)</mo></math></span>, we study the linear code <span><math><mi>C</mi><mo>(</mo><msub><mrow><mi>Λ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></math></span> arising from it. We show that <span><math><mi>C</mi><mo>(</mo><msub><mrow><mi>Λ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></math></span> is a minimal code and we determine its basic parameters, its full weight list and its linear automorphism group. We also give a geometrical characterization of its minimum and second lowest weight codewords as well as of some of the words of maximum weight.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"111 ","pages":"Article 102766"},"PeriodicalIF":1.2,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145684470","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}
Pub Date : 2026-03-01Epub Date: 2025-12-30DOI: 10.1016/j.ffa.2025.102788
Usman Mushrraf, Ferdinando Zullo
One-weight codes, in which all nonzero codewords share the same weight, form a highly structured class of linear codes with deep connections to finite geometry. While their classification is well understood in the Hamming and rank metrics—being equivalent to (direct sums of) simplex codes—the sum-rank metric presents a far more intricate landscape. In this work, we explore the geometry of one-weight sum-rank metric codes, focusing on three distinct classes. First, we introduce and classify constant rank-list sum-rank metric codes, where each nonzero codeword has the same tuple of ranks, extending results from the rank-metric setting. Next, we investigate the more general constant rank-profile codes, where, up to reordering, each nonzero codeword has the same tuple of ranks. Although a complete classification remains elusive, we present the first examples and partial structural results for this class. Finally, we consider one-weight codes that are also MSRD (Maximum Sum-Rank Distance) codes. For dimension two, constructions arise from partitions of scattered linear sets on projective lines. For dimension three, we connect their existence to that of special 2-fold blocking sets in the projective plane, leading to new bounds and nonexistence results over certain fields.
{"title":"One-weight codes in the sum-rank metric","authors":"Usman Mushrraf, Ferdinando Zullo","doi":"10.1016/j.ffa.2025.102788","DOIUrl":"10.1016/j.ffa.2025.102788","url":null,"abstract":"<div><div>One-weight codes, in which all nonzero codewords share the same weight, form a highly structured class of linear codes with deep connections to finite geometry. While their classification is well understood in the Hamming and rank metrics—being equivalent to (direct sums of) simplex codes—the sum-rank metric presents a far more intricate landscape. In this work, we explore the geometry of one-weight sum-rank metric codes, focusing on three distinct classes. First, we introduce and classify <em>constant rank-list</em> sum-rank metric codes, where each nonzero codeword has the same tuple of ranks, extending results from the rank-metric setting. Next, we investigate the more general <em>constant rank-profile</em> codes, where, up to reordering, each nonzero codeword has the same tuple of ranks. Although a complete classification remains elusive, we present the first examples and partial structural results for this class. Finally, we consider one-weight codes that are also MSRD (Maximum Sum-Rank Distance) codes. For dimension two, constructions arise from partitions of scattered linear sets on projective lines. For dimension three, we connect their existence to that of special 2-fold blocking sets in the projective plane, leading to new bounds and nonexistence results over certain fields.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"111 ","pages":"Article 102788"},"PeriodicalIF":1.2,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145883542","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}
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