Finite Fields and Their Applications最新文献

{"title":"Transverse-freeness in finite geometries","authors":"Charlie Bruggemann ,&nbsp;Vera Choi ,&nbsp;Brian Freidin ,&nbsp;Jaedon Whyte","doi":"10.1016/j.ffa.2025.102663","DOIUrl":"10.1016/j.ffa.2025.102663","url":null,"abstract":"<div><div>We study the interplay between combinatorial and algebraic geometry via projective curves and hypersurfaces defined over a finite field that are tangent to every member of a class of low-degree varieties. Extending the 2-dimensional work of Asgarli, we first explore the lowest degrees attainable by smooth hypersurfaces in <em>n</em>-dimensional projective space that are tangent to every <em>k</em>-dimensional subspace, for some values of <em>n</em> and <em>k</em>. We then study projective surfaces that serve as models of finite inversive and hyperbolic planes, finite analogs of spherical and hyperbolic geometries. In these surfaces, we prove existence results for low-degree curves tangent to each of the lowest degree curves defined over the base field.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"108 ","pages":"Article 102663"},"PeriodicalIF":1.2,"publicationDate":"2025-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144203078","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":"2025-06-02","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":"Two families of self-orthogonal codes with applications in LCD codes and optimally extendable codes","authors":"Dengcheng Xie,&nbsp;Shixin Zhu","doi":"10.1016/j.ffa.2025.102659","DOIUrl":"10.1016/j.ffa.2025.102659","url":null,"abstract":"<div><div>Self-orthogonal codes have interesting applications in quantum codes, linear complementary dual (LCD) codes and lattices. LCD codes and (almost) optimally extendable codes are useful to safeguard against Side-Channel Attacks (SCAs) and Fault Injection Attacks (FIAs). In this paper, we first give a lower bound of dual distances for augmented codes via the defining-set construction. Then we construct two families of <em>q</em>-ary self-orthogonal codes with determined weight distributions via defining-set construction and propose the parameters of their duals. Besides, several families of AMDS codes are obtained as byproducts, which are both length-optimal and dimension-optimal with respect to the Sphere-packing bound. As applications, these self-orthogonal codes are used to construct LCD codes and proved to be optimally extendable. As a consequence, our constructions produce some optimal codes.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"108 ","pages":"Article 102659"},"PeriodicalIF":1.2,"publicationDate":"2025-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144167720","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":"Almost Steiner systems in finite classical polar spaces","authors":"Yunxian Wu ,&nbsp;Tao Feng ,&nbsp;Lei Xu ,&nbsp;Menglong Zhang","doi":"10.1016/j.ffa.2025.102662","DOIUrl":"10.1016/j.ffa.2025.102662","url":null,"abstract":"<div><div>Let <span><math><mi>Q</mi></math></span> be a finite classical polar space with rank <em>n</em> and <span><math><mi>P</mi></math></span> be a collection of <em>k</em>-dimensional subspaces in <span><math><mi>Q</mi></math></span> called blocks. Let Λ be a set of nonnegative integers. A pair <span><math><mo>(</mo><mi>Q</mi><mo>,</mo><mi>P</mi><mo>)</mo></math></span> is called a <em>t</em>-<span><math><msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>Λ</mi><mo>)</mo></mrow><mrow><mi>Q</mi></mrow></msub></math></span> design if every <em>t</em>-dimensional subspace in <span><math><mi>Q</mi></math></span> is contained in exactly <span><math><mi>λ</mi><mo>∈</mo><mi>Λ</mi></math></span> blocks of <span><math><mi>P</mi></math></span>. A <em>t</em>-<span><math><msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>,</mo><mn>1</mn><mo>)</mo></mrow><mrow><mi>Q</mi></mrow></msub></math></span> design is called a <em>Q</em>-Steiner system, which has <span><math><msub><mrow><mo>[</mo><mtable><mtr><mtd><mi>n</mi></mtd></mtr><mtr><mtd><mi>t</mi></mtd></mtr></mtable><mo>]</mo></mrow><mrow><mi>Q</mi></mrow></msub><mo>/</mo><msub><mrow><mo>[</mo><mtable><mtr><mtd><mi>k</mi></mtd></mtr><mtr><mtd><mi>t</mi></mtd></mtr></mtable><mo>]</mo></mrow><mrow><mi>q</mi></mrow></msub></math></span> blocks. Despite knowing very little about the existence of <em>Q</em>-Steiner systems, we demonstrate that given any positive integers <em>k</em> and <em>t</em> satisfying <span><math><mi>k</mi><mo>&gt;</mo><mi>t</mi></math></span>, for any finite polar space <span><math><mi>Q</mi></math></span> with a sufficiently large rank <em>n</em>, there exists a <em>t</em>-<span><math><msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>,</mo><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>}</mo><mo>)</mo></mrow><mrow><mi>Q</mi></mrow></msub></math></span> design with <span><math><mo>(</mo><mn>1</mn><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>)</mo><msub><mrow><mo>[</mo><mtable><mtr><mtd><mi>n</mi></mtd></mtr><mtr><mtd><mi>t</mi></mtd></mtr></mtable><mo>]</mo></mrow><mrow><mi>Q</mi></mrow></msub><mo>/</mo><msub><mrow><mo>[</mo><mtable><mtr><mtd><mi>k</mi></mtd></mtr><mtr><mtd><mi>t</mi></mtd></mtr></mtable><mo>]</mo></mrow><mrow><mi>q</mi></mrow></msub></math></span> blocks.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"108 ","pages":"Article 102662"},"PeriodicalIF":1.2,"publicationDate":"2025-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144167721","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":"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}