Finite Fields and Their Applications最新文献

{"title":"De Bruijn tori without zeros: a field-theoretic perspective","authors":"Ming Hsuan Kang,&nbsp;Yu Hsuan Hsieh","doi":"10.1016/j.ffa.2026.102790","DOIUrl":"10.1016/j.ffa.2026.102790","url":null,"abstract":"<div><div>We present an algebraic construction of trace-based De Bruijn tori over finite fields, focusing on the nonzero variant that omits the all-zero pattern. The construction arranges nonzero field elements on a toroidal grid using two multiplicatively independent generators, with values obtained by applying a fixed linear map, typically the field trace.</div><div>We characterize sampling patterns as subsets whose associated field elements form an <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>-basis, and show that column structures correspond to cyclic shifts of De Bruijn sequences determined by irreducible polynomials over subfields. Recursive update rules based on multiplicative translations enable efficient computation.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"112 ","pages":"Article 102790"},"PeriodicalIF":1.2,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145981218","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":"Quasi-optimum distance flag codes","authors":"Clementa Alonso-González,&nbsp;Miguel Ángel Navarro-Pérez","doi":"10.1016/j.ffa.2026.102799","DOIUrl":"10.1016/j.ffa.2026.102799","url":null,"abstract":"<div><div>A <em>flag</em> is a sequence of nested subspaces of a given ambient space <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> over a finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>. In network coding, a <em>flag code</em> is a set of flags, all of them with the same sequence of dimensions, the <em>type vector</em>. In this paper, we investigate <em>quasi-optimum distance flag codes</em>, i.e., those attaining the second best possible distance value. We characterize them and present upper bounds for their cardinality. Moreover, we propose a systematic construction for every choice of the type vector by using <em>partial spreads</em> and <em>sunflowers</em>. For flag codes with lower minimum distance, we adapt the previous construction and provide some results towards their characterization, especially in the case of the third best possible distance value.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"112 ","pages":"Article 102799"},"PeriodicalIF":1.2,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145981220","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":"Counterexamples, constructions, and nonexistence results for optimal ternary cyclic codes","authors":"Jingjun Bao ,&nbsp;Hanlin Zou","doi":"10.1016/j.ffa.2026.102795","DOIUrl":"10.1016/j.ffa.2026.102795","url":null,"abstract":"<div><div>Cyclic codes are an important subclass of linear codes with wide applications in communication systems and data storage systems. In 2013, Ding and Helleseth presented nine open problems on optimal ternary cyclic codes <span><math><msub><mrow><mi>C</mi></mrow><mrow><mo>(</mo><mn>1</mn><mo>,</mo><mi>e</mi><mo>)</mo></mrow></msub></math></span>. While the first two and the sixth problems have been fully solved, others remain open. In this paper, we advance the study of the third and fourth open problems by providing the first counterexamples to both and constructing two families of optimal codes under certain conditions, thereby partially solving the third problem. Furthermore, we investigate the cyclic codes <span><math><msub><mrow><mi>C</mi></mrow><mrow><mo>(</mo><mn>1</mn><mo>,</mo><mi>e</mi><mo>)</mo></mrow></msub></math></span> where <span><math><mi>e</mi><mo>(</mo><msup><mrow><mn>3</mn></mrow><mrow><mi>h</mi></mrow></msup><mo>±</mo><mn>1</mn><mo>)</mo><mo>≡</mo><mfrac><mrow><msup><mrow><mn>3</mn></mrow><mrow><mi>m</mi></mrow></msup><mo>−</mo><mi>a</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><msup><mrow><mn>3</mn></mrow><mrow><mi>m</mi></mrow></msup><mo>−</mo><mn>1</mn><mo>)</mo></math></span> and <em>a</em> is odd. For <span><math><mi>a</mi><mo>≡</mo><mn>3</mn><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mn>4</mn><mo>)</mo></math></span>, we present two new families of optimal codes with parameters <span><math><mo>[</mo><msup><mrow><mn>3</mn></mrow><mrow><mi>m</mi></mrow></msup><mo>−</mo><mn>1</mn><mo>,</mo><msup><mrow><mn>3</mn></mrow><mrow><mi>m</mi></mrow></msup><mo>−</mo><mn>1</mn><mo>−</mo><mn>2</mn><mi>m</mi><mo>,</mo><mn>4</mn><mo>]</mo></math></span>, generalizing known constructions. For <span><math><mi>a</mi><mo>≡</mo><mn>1</mn><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mn>4</mn><mo>)</mo></math></span>, we obtain several nonexistence results on optimal codes <span><math><msub><mrow><mi>C</mi></mrow><mrow><mo>(</mo><mn>1</mn><mo>,</mo><mi>e</mi><mo>)</mo></mrow></msub></math></span> with the aforementioned parameters revealing the constraints of such codes.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"112 ","pages":"Article 102795"},"PeriodicalIF":1.2,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146039254","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":"Function-correcting codes with homogeneous distance","authors":"Huiying Liu,&nbsp;Hongwei Liu","doi":"10.1016/j.ffa.2026.102791","DOIUrl":"10.1016/j.ffa.2026.102791","url":null,"abstract":"<div><div>Function-correcting codes are designed to reduce redundancy of codes when protecting function values of information against errors. As generalizations of Hamming weights and Lee weights over <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>, homogeneous weights are used in codes over finite rings. In this paper, we introduce function-correcting codes with homogeneous distance denoted by FCCHDs, which extend function-correcting codes with Hamming distance. We first define <em>D</em>-homogeneous distance codes. We use <em>D</em>-homogeneous distance codes to characterize connections between the optimal redundancy of FCCHDs and lengths of these codes for some matrices <em>D</em>. By these connections, we obtain several bounds of the optimal redundancy of FCCHDs for some functions. In addition, we also construct FCCHDs for homogeneous weight functions and homogeneous weight distribution functions. Specially, redundancies of some codes we construct in this paper reach the optimal redundancy bounds.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"112 ","pages":"Article 102791"},"PeriodicalIF":1.2,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145950123","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":"Easy repair via codes with simplex locality","authors":"M. Kuijper ,&nbsp;J. Lieb ,&nbsp;D. Napp","doi":"10.1016/j.ffa.2026.102809","DOIUrl":"10.1016/j.ffa.2026.102809","url":null,"abstract":"<div><div>In the context of distributed storage systems, locally repairable codes have become important. In this paper we focus on codes that allow for multi-erasure pattern decoding with low computational effort. Different optimality requirements, measured by the code's rate, minimum distance, locality, availability as well as field size, influence each other and can not all be maximized at the same time. We focus on the notion of easy repair, more specifically on the construction of codes that can repair correctable erasure patterns with minimal computational effort. In particular, we introduce the easy repair property and then present binary codes of different rates that possess this property. The presented codes are all in some way related to binary simplex codes and include block codes as well as unit-memory convolutional codes. We also formulate conditions under which the easy repairs can be performed in parallel, thus improving access speed of the distributed storage system.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"112 ","pages":"Article 102809"},"PeriodicalIF":1.2,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146189256","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":"Sums of two squares over Fq[T]","authors":"Wentang Kuo,&nbsp;Yu-Ru Liu,&nbsp;Yash Totani","doi":"10.1016/j.ffa.2026.102812","DOIUrl":"10.1016/j.ffa.2026.102812","url":null,"abstract":"<div><div>In this paper, we establish a formula for the number of representations of a polynomial as a norm from a quadratic extension over function fields and study its moments. Our approach involves employing two distinct techniques to derive the main results concerning asymptotic formulas for the moments. The first technique utilizes the framework of Dirichlet series and the second technique involves effectively partitioning the set of polynomials of a fixed degree, providing asymptotic formulas in the limit of large polynomial degree.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"112 ","pages":"Article 102812"},"PeriodicalIF":1.2,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146188086","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 the form (xq−x+δ)i(q−1)+1+L(x) over Fq2","authors":"Rohit Gupta ,&nbsp;Amritanshu Rai","doi":"10.1016/j.ffa.2026.102797","DOIUrl":"10.1016/j.ffa.2026.102797","url":null,"abstract":"<div><div>Let <em>q</em> be a power of a prime number and let <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> be the finite field with <em>q</em> elements. Let <span><math><mi>δ</mi><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub></math></span> be arbitrary. In this paper, we give a relationship between the permutation property of polynomials over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub></math></span> of the forms <span><math><mi>g</mi><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>q</mi></mrow></msup><mo>−</mo><mi>x</mi><mo>+</mo><mi>δ</mi><mo>)</mo><mo>+</mo><mi>c</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>q</mi></mrow></msup><mo>+</mo><mi>d</mi><mi>x</mi></math></span> and <span><math><mi>g</mi><msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mrow><mi>q</mi></mrow></msup><mo>−</mo><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>+</mo><mi>c</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>q</mi></mrow></msup><mo>+</mo><mi>d</mi><mi>x</mi></math></span> where <span><math><mi>c</mi><mo>,</mo><mi>d</mi><mo>∈</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup></math></span>, <span><math><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub><mo>[</mo><mi>x</mi><mo>]</mo></math></span>. Further, we find the necessary and sufficient conditions on the coefficients <em>c</em> and <em>d</em> such that polynomials of the forms <span><math><msup><mrow><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>q</mi></mrow></msup><mo>−</mo><mi>x</mi><mo>+</mo><mi>δ</mi><mo>)</mo></mrow><mrow><mi>i</mi><mo>(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>+</mo><mn>1</mn></mrow></msup><mo>+</mo><mi>c</mi><mi>x</mi></math></span> and <span><math><msup><mrow><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>q</mi></mrow></msup><mo>−</mo><mi>x</mi><mo>+</mo><mi>δ</mi><mo>)</mo></mrow><mrow><mi>i</mi><mo>(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>+</mo><mn>1</mn></mrow></msup><mo>+</mo><mi>c</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>q</mi></mrow></msup><mo>+</mo><mi>d</mi><mi>x</mi></math></span> permute <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub></math></span>. Moreover, some results of this article supersede certain results in the related literature.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"112 ","pages":"Article 102797"},"PeriodicalIF":1.2,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145981222","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":"Hermitian-Singer functional and differential codes","authors":"Gábor Korchmáros ,&nbsp;Federico Romaniello ,&nbsp;Valentino Smaldore","doi":"10.1016/j.ffa.2026.102794","DOIUrl":"10.1016/j.ffa.2026.102794","url":null,"abstract":"<div><div>Algebraic geometry codes on the Hermitian curve have been the subject of several papers, since they happen to have good performances and large automorphism groups. Here, those arising from the Singer cycle of the Hermitian curve are investigated.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"112 ","pages":"Article 102794"},"PeriodicalIF":1.2,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145981221","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 index q + 1 over Fq2","authors":"Xiutao Feng ,&nbsp;Qiang Wang","doi":"10.1016/j.ffa.2026.102796","DOIUrl":"10.1016/j.ffa.2026.102796","url":null,"abstract":"<div><div>We provide a generic construction of permutation polynomials over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub></math></span> with index <span><math><mi>q</mi><mo>+</mo><mn>1</mn></math></span> from any permutation polynomial of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>. We also extend our construction using polynomials with coefficients in <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub></math></span> such that they are injective over a subset of <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mo>⁎</mo></mrow></msubsup></math></span>, which corresponds to the set <span><math><msub><mrow><mi>μ</mi></mrow><mrow><mi>q</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span> of all <span><math><mo>(</mo><mi>q</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>-th roots of unity.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"112 ","pages":"Article 102796"},"PeriodicalIF":1.2,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146039253","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":"Twisted group algebra of dihedral groups over finite fields","authors":"André Duarte","doi":"10.1016/j.ffa.2026.102792","DOIUrl":"10.1016/j.ffa.2026.102792","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> be a finite field and <span><math><msub><mrow><mi>D</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is the dihedral group of order 2<em>n</em>. We present formulas for a complete set of 2-cocycles of <span><math><msub><mrow><mi>D</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> and compute the primitive central idempotents of <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>α</mi></mrow></msubsup><msub><mrow><mi>D</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. We conclude by describing the Wedderburn decomposition of <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>α</mi></mrow></msubsup><msub><mrow><mi>D</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> and the irreducible projective representations of <span><math><msub><mrow><mi>D</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"112 ","pages":"Article 102792"},"PeriodicalIF":1.2,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145950124","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}