Finite Fields and Their Applications最新文献

{"title":"Linear codes with few weights from vectorial dual-bent functions","authors":"Zhicheng Wang ,&nbsp;Qiang Wang ,&nbsp;Shudi Yang","doi":"10.1016/j.ffa.2025.102660","DOIUrl":"10.1016/j.ffa.2025.102660","url":null,"abstract":"<div><div>Linear codes with few weights have wide applications in secret sharing, authentication codes, strongly regular graphs and association schemes. In this paper, we present linear codes from vectorial dual-bent functions and permutation polynomials, such that their parameters and weight distributions can be explicitly determined. In particular, some of them are three-weight optimal or almost optimal codes. As applications, we extend these codes to construct self-orthogonal codes and show the existence of asymmetric quantum codes.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"108 ","pages":"Article 102660"},"PeriodicalIF":1.2,"publicationDate":"2025-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144154384","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":"Shortest-path and antichain metrics","authors":"Mahir Bilen Can,&nbsp;Dillon Montero","doi":"10.1016/j.ffa.2025.102658","DOIUrl":"10.1016/j.ffa.2025.102658","url":null,"abstract":"<div><div>In this paper, we introduce two new metrics for error-correcting codes that extend the classical Hamming metric. The first, called the shortest-path metric, coincides with the Niederreiter-Rosenbloom-Tsfasman (NRT) metric when the underlying poset is a disjoint union of equal-length chains. The second, called the antichain metric, is shown to align with the <em>b</em>-symbol Hamming weight under the same poset structure. We explore analogs of maximum distance separable (MDS) codes and perfect codes for both metrics and determine the corresponding weight enumerator polynomials. Additionally, we establish criteria for when the antichain metric yields one-weight perfect codes.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"108 ","pages":"Article 102658"},"PeriodicalIF":1.2,"publicationDate":"2025-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144134401","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 distribution of prime independent multiplicative functions over function fields","authors":"Matilde Lalín ,&nbsp;Olha Zhur","doi":"10.1016/j.ffa.2025.102657","DOIUrl":"10.1016/j.ffa.2025.102657","url":null,"abstract":"<div><div>We consider the family of multiplicative functions of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><mi>T</mi><mo>]</mo></math></span> with the property that the value at a power of an irreducible polynomial depends only on the exponent, but does not depend on the polynomial or its degree. We study variances of such functions in various regimes, relating them to variances of the divisor function <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>f</mi><mo>)</mo></math></span>. We examine different settings that can be related to distributions over the ensembles of unitary matrices, symplectic matrices, and orthogonal matrices as in the works of <span><span>[18]</span></span>, <span><span>[19]</span></span>, <span><span>[20]</span></span>. While most questions give very similar answers as the distributions of the divisor function, some of the symplectic problems, dealing with quadratic characters, are different and vary according to the values of the function at the square of the primes.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"107 ","pages":"Article 102657"},"PeriodicalIF":1.2,"publicationDate":"2025-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144090381","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":"Computing the minimum distance of the C(O3,6) polar orthogonal Grassmann code with elementary methods","authors":"Sarah Gregory ,&nbsp;Fernando Piñero González ,&nbsp;Doel Rivera–Laboy ,&nbsp;Lani Southern","doi":"10.1016/j.ffa.2025.102656","DOIUrl":"10.1016/j.ffa.2025.102656","url":null,"abstract":"<div><div>The polar orthogonal Grassmann code <span><math><mi>C</mi><mo>(</mo><msub><mrow><mi>O</mi></mrow><mrow><mn>3</mn><mo>,</mo><mn>6</mn></mrow></msub><mo>)</mo></math></span> is the linear code associated to the polar orthogonal Grassmannian subvariety of the Grassmannian. The variety <span><math><msub><mrow><mi>O</mi></mrow><mrow><mn>3</mn><mo>,</mo><mn>6</mn></mrow></msub></math></span> is the Grassmannian of 3-spaces contained in a hyperbolic quadric in <span><math><mi>P</mi><mi>G</mi><mo>(</mo><mn>6</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span>. In this manuscript we prove that the minimum distance of the polar orthogonal Grassmann code <span><math><mi>C</mi><mo>(</mo><msub><mrow><mi>O</mi></mrow><mrow><mn>3</mn><mo>,</mo><mn>6</mn></mrow></msub><mo>)</mo></math></span> is <span><math><msup><mrow><mi>q</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>−</mo><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> for <em>q</em> odd and <span><math><msup><mrow><mi>q</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> for <em>q</em> even. We also prove that the minimum distance of <span><math><mi>C</mi><mo>(</mo><msub><mrow><mi>O</mi></mrow><mrow><mn>4</mn><mo>,</mo><mn>8</mn></mrow></msub><mo>)</mo></math></span> is <span><math><msup><mrow><mi>q</mi></mrow><mrow><mn>6</mn></mrow></msup></math></span> when <em>q</em> is even. Our technique is based on partitioning <span><math><msub><mrow><mi>O</mi></mrow><mrow><mn>3</mn><mo>,</mo><mn>6</mn></mrow></msub></math></span> into different sets such that on each partition the code <span><math><mi>C</mi><mo>(</mo><msub><mrow><mi>O</mi></mrow><mrow><mn>3</mn><mo>,</mo><mn>6</mn></mrow></msub><mo>)</mo></math></span> is identified with evaluations of determinants of skew–symmetric matrices. Our bounds come from elementary algebraic methods counting the zeroes of particular classes of polynomials. The techniques presented in this paper may be adapted for other polar Grassmannians.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"107 ","pages":"Article 102656"},"PeriodicalIF":1.2,"publicationDate":"2025-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143940684","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":"Simple matrix cryptosystem is not broken by Liu's attack","authors":"Lih-Chung Wang,&nbsp;Yen-Liang Kuan,&nbsp;Po-En Tseng,&nbsp;Chun-Yen Chou","doi":"10.1016/j.ffa.2025.102643","DOIUrl":"10.1016/j.ffa.2025.102643","url":null,"abstract":"<div><div>At PQCrypto2013, Tao et al. proposed a new multivariate public key cryptosystem for encryption called Simple Matrix (or ABC) encryption scheme. In 2018, Liu et al. proposed a key recovery attack on ABC scheme with claimed complexity of <span><math><mi>O</mi><mrow><mo>(</mo><msup><mrow><mi>s</mi></mrow><mrow><mn>4</mn><mi>w</mi></mrow></msup><mo>)</mo></mrow></math></span>, where <em>s</em> is the size of the <span><math><mi>s</mi><mo>×</mo><mi>s</mi></math></span> square matrices in the scheme, <span><math><mi>w</mi><mo>=</mo><mn>3</mn></math></span> in the usual Gaussian elimination algorithm and <span><math><mi>w</mi><mo>=</mo><mn>2.3776</mn></math></span> in improved scheme. In this paper, we show that Liu's attack only works for the case <span><math><mi>s</mi><mo>=</mo><mn>2</mn></math></span> of ABC scheme which means that Liu's attack doesn't break ABC scheme.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"107 ","pages":"Article 102643"},"PeriodicalIF":1.2,"publicationDate":"2025-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143928788","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":"Several new classes of p-ary weakly regular plateaued functions and minimal codes with several weights","authors":"Wengang Jin,&nbsp;Kangquan Li,&nbsp;Longjiang Qu","doi":"10.1016/j.ffa.2025.102644","DOIUrl":"10.1016/j.ffa.2025.102644","url":null,"abstract":"<div><div>Plateaued functions, including bent functions, are crucial in cryptography due to their possession of a range of desirable cryptographic properties. Weakly regular plateaued functions can also be employed in many domains. In particular, they have been widely used in designing good linear codes for several applications (such as secret sharing and two-party computation), association schemes, and strongly regular graphs. This paper is devoted to weakly regular plateaued functions, whose objectives are twofold. First, we aim to generate new infinite families of weakly regular plateaued functions and then, to design new families of <em>p</em>-ary linear codes and investigate their use for some standard applications after studying its minimality based on their weight distributions. More specifically, we present several classes of weakly regular plateaued functions from monomial bent functions, and determine their corresponding dual functions explicitly. Furthermore, we exploit our constructions to derive several new classes of minimal linear codes violating the Ashikhmin-Barg condition with six, seven, nine, ten or eleven weights, which are more appropriate for several applications.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"107 ","pages":"Article 102644"},"PeriodicalIF":1.2,"publicationDate":"2025-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143922453","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":"2025-05-08","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":"Products of involutions in symplectic groups over general fields (II): Finite fields","authors":"Clément de Seguins Pazzis","doi":"10.1016/j.ffa.2025.102641","DOIUrl":"10.1016/j.ffa.2025.102641","url":null,"abstract":"<div><div>Let <em>s</em> be an <em>n</em>-dimensional symplectic form over a field <span><math><mi>F</mi></math></span> of characteristic other than 2, with <span><math><mi>n</mi><mo>&gt;</mo><mn>2</mn></math></span>.</div><div>In a previous article, we have proved that if <span><math><mi>F</mi></math></span> is infinite then every element of the symplectic group <span><math><mi>Sp</mi><mo>(</mo><mi>s</mi><mo>)</mo></math></span> is the product of four involutions if <em>n</em> is a multiple of 4 and of five involutions otherwise.</div><div>Here, we adapt this result to all finite fields with characteristic not 2, with the sole exception of the very special situation where <span><math><mi>n</mi><mo>=</mo><mn>4</mn></math></span> and <span><math><mo>|</mo><mi>F</mi><mo>|</mo><mo>=</mo><mn>3</mn></math></span>, a special case which we study extensively.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"107 ","pages":"Article 102641"},"PeriodicalIF":1.2,"publicationDate":"2025-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143918444","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 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}