Finite Fields and Their Applications最新文献

Weierstrass semigroups and automorphism group of a maximal function field with the third largest possible genus, q ≡ 1 (mod 3) 具有第三大可能属q的极大函数域的Weierstrass半群和自同构群 ≡ 1 （mod 3）

IF 1.2 3区数学

Finite Fields and Their Applications Pub Date : 2025-07-21 DOI: 10.1016/j.ffa.2025.102701

Peter Beelen , Maria Montanucci , Lara Vicino

{"title":"Weierstrass semigroups and automorphism group of a maximal function field with the third largest possible genus, q ≡ 1 (mod 3)","authors":"Peter Beelen , Maria Montanucci , Lara Vicino","doi":"10.1016/j.ffa.2025.102701","DOIUrl":"10.1016/j.ffa.2025.102701","url":null,"abstract":"<div><div>In this article we continue the work started in [3], explicitly determining the Weierstrass semigroup at any place and the full automorphism group of a known <math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub></math>-maximal function field <math><msub><mrow><mi>Y</mi></mrow><mrow><mn>3</mn></mrow></msub></math> having the third largest genus, for <math><mi>q</mi><mo>≡</mo><mn>1</mn><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mn>3</mn><mo>)</mo></math>. This function field arises as a Galois subfield of the Hermitian function field, and its uniqueness (with respect to the value of its genus) is a well-known open problem. Knowing the Weierstrass semigroups may provide a key towards solving this problem. Surprisingly enough, <math><msub><mrow><mi>Y</mi></mrow><mrow><mn>3</mn></mrow></msub></math> has many different types of Weierstrass semigroups and the set of its Weierstrass places is much richer than its set of <math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub></math>-rational places. We show that a similar exceptional behaviour does not occur in terms of automorphisms, that is, <math><mrow><mi>Aut</mi></mrow><mo>(</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>)</mo></math> is exactly the automorphism group inherited from the Hermitian function field, apart from small values of q.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"109 ","pages":"Article 102701"},"PeriodicalIF":1.2,"publicationDate":"2025-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144670374","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}

引用次数: 0

Corrigendum to “Higher Grassmann codes II” [Finite Fields Appl. 89 (2023) 102211] “高等格拉斯曼规范II”的勘误表[有限域应用89 (2023)102211]

IF 1.2 3区数学

Finite Fields and Their Applications Pub Date : 2025-07-17 DOI: 10.1016/j.ffa.2025.102700

Mahir Bilen Can , Roy Joshua , G.V. Ravindra

引用次数: 0

On iso-dual MDS codes from elliptic curves 椭圆曲线上的等对偶MDS码

IF 1.2 3区数学

Finite Fields and Their Applications Pub Date : 2025-07-16 DOI: 10.1016/j.ffa.2025.102699

Yunlong Zhu , Chang-An Zhao

{"title":"On iso-dual MDS codes from elliptic curves","authors":"Yunlong Zhu , Chang-An Zhao","doi":"10.1016/j.ffa.2025.102699","DOIUrl":"10.1016/j.ffa.2025.102699","url":null,"abstract":"<div><div>For a linear code C over a finite field, if its dual code <math><msup><mrow><mi>C</mi></mrow><mrow><mo>⊥</mo></mrow></msup></math> is equivalent to itself, then the code C is said to be isometry-dual. In this paper, we first confirm a conjecture about the isometry-dual MDS elliptic codes proposed by Han and Ren. Subsequently, two constructions of isometry-dual maximum distance separable (MDS) codes from elliptic curves are presented. The new code length n satisfies <math><mi>n</mi><mo>≤</mo><mfrac><mrow><mi>q</mi><mo>+</mo><mo>⌊</mo><mn>2</mn><msqrt><mrow><mi>q</mi></mrow></msqrt><mo>⌋</mo><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math> when q is even and <math><mi>n</mi><mo>≤</mo><mfrac><mrow><mi>q</mi><mo>+</mo><mo>⌊</mo><mn>2</mn><msqrt><mrow><mi>q</mi></mrow></msqrt><mo>⌋</mo><mo>−</mo><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math> when q is odd. Additionally, we consider the hull dimension of both constructions. In the case of finite fields with even characteristics, an isometry-dual MDS code is equivalent to a self-dual MDS code and a linear complementary dual MDS code. Finally, we apply our results to entanglement-assisted quantum error correcting codes (EAQECCs) and obtain two new families of MDS EAQECCs.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"109 ","pages":"Article 102699"},"PeriodicalIF":1.2,"publicationDate":"2025-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144632513","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}

引用次数: 0

Evaluations of Eisenstein sums 爱森斯坦和的求值

IF 1.2 3区数学

Finite Fields and Their Applications Pub Date : 2025-07-11 DOI: 10.1016/j.ffa.2025.102698

Ron Evans , Mark Van Veen

引用次数: 0

Square values of several polynomials over a finite field 有限域上若干多项式的平方值

IF 1.2 3区数学

Finite Fields and Their Applications Pub Date : 2025-07-11 DOI: 10.1016/j.ffa.2025.102696

Kaloyan Slavov

引用次数: 0

Closed formulas for the generators of all constacyclic codes and for the factorization of Xn − 1, the n-th cyclotomic polynomial and every composition of the form f(Xn) over a finite field for arbitrary positive integers n 所有常环码的生成和Xn的分解的封闭公式 − 1，第n个环多项式和任意正整数n在有限域上的形式f（Xn）的每一个组合

IF 1.2 3区数学

Finite Fields and Their Applications Pub Date : 2025-07-11 DOI: 10.1016/j.ffa.2025.102695

Anna-Maurin Graner

{"title":"Closed formulas for the generators of all constacyclic codes and for the factorization of Xn − 1, the n-th cyclotomic polynomial and every composition of the form f(Xn) over a finite field for arbitrary positive integers n","authors":"Anna-Maurin Graner","doi":"10.1016/j.ffa.2025.102695","DOIUrl":"10.1016/j.ffa.2025.102695","url":null,"abstract":"<div><div>In this paper we present the solution to four 150 year-old closely related problems in the study of polynomials and algebraic codes over a finite field <math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math>. We give one closed formula each for the factorization of the polynomials <math><msup><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>−</mo><mi>a</mi></math> for arbitrary <math><mi>a</mi><mo>∈</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup></math>, <math><msup><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>−</mo><mn>1</mn></math>, the n-th cyclotomic polynomial and the composition <math><mi>f</mi><mo>(</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></math> for arbitrary monic irreducible polynomials <math><mi>f</mi><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><mi>X</mi><mo>]</mo></math>, <math><mi>f</mi><mo>≠</mo><mi>X</mi></math>, for arbitrary positive integers n. With a new perspective on these problems we show that the factorization of <math><msup><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>−</mo><mi>a</mi></math> has a beautiful underlying structure which is completely determined by the order of a in the multiplicative group <math><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup></math>.</div><div>The binomial <math><msup><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>−</mo><mi>a</mi></math> and the composition <math><mi>f</mi><mo>(</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></math> were first considered over prime fields by Joseph Alfred Serret in 1866. In recent years, the factorization of <math><msup><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>−</mo><mi>a</mi></math> has been studied extensively due to the fact that its factors are the generators of the popular constacyclic codes over finite fields. The factorization of <math><msup><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>−</mo><mn>1</mn></math> and of its famous factor, the n-th cyclotomic polynomial, was first studied over prime fields by Carl Friedrich Gauss (among others) in the middle of the 19th century. Since then, many mathematicians were fascinated by this factorization and nowadays it is needed for the construction of the cyclic codes over finite fields.</div><div>Many formulas for the factorization of the four polynomials for special choices of n and a were obtained by a large number of mathematicians. Our formulas naturally extend, cover, combine and complete all of these partial solutions.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"109 ","pages":"Article 102695"},"PeriodicalIF":1.2,"publicationDate":"2025-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144596934","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}

引用次数: 0

Bounds on the size of (r,δ)-locally repairable codes for fixed values q and d （r,δ）-定值q和d的局部可修码的大小界限

IF 1.2 3区数学

Finite Fields and Their Applications Pub Date : 2025-07-11 DOI: 10.1016/j.ffa.2025.102697

Wenqin Zhang , Chen Yuan , Yuan Luo , Nian Li

{"title":"Bounds on the size of (r,δ)-locally repairable codes for fixed values q and d","authors":"Wenqin Zhang , Chen Yuan , Yuan Luo , Nian Li","doi":"10.1016/j.ffa.2025.102697","DOIUrl":"10.1016/j.ffa.2025.102697","url":null,"abstract":"<div><div>Erasure codes can strengthen fault-tolerance and reliability in distributed storage systems. One of these, locally repairable codes (LRCs), plays a crucial role. A locally repairable code with locality r (called r-LRC) can recover any coded symbol by accessing at most r other coded symbols. The original definition of LRCs is used to repair a single failed node. To address the practical concern of multiple node failures, the concept of LRCs with locality <math><mo>(</mo><mi>r</mi><mo>,</mo><mi>δ</mi><mo>)</mo></math> was introduced by Prakash et al. (2012) which can be seen as a generalization of r-LRCs. Since then, the bounds and constructions of <math><mo>(</mo><mi>r</mi><mo>,</mo><mi>δ</mi><mo>)</mo></math>-LRCs have been extensively studied.</div><div>This paper is dedicated to both the bounds and constructions of <math><mo>(</mo><mi>r</mi><mo>,</mo><mi>δ</mi><mo>)</mo></math>-LRCs with disjoint local repair groups over a finite field <math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math>, where the parameters r, δ, the alphabet size q, and the minimum distance d are fixed constants, while the code length n tends to infinity. Inspired by the method of the classical Gilbert-Varshamov (GV) bound, we first derive an asymptotic Gilbert-Varshamov-type bound for <math><mo>(</mo><mi>r</mi><mo>,</mo><mi>δ</mi><mo>)</mo></math>-LRCs in this regime. We manage to show that this GV-type bound works as a threshold for random linear <math><mo>(</mo><mi>r</mi><mo>,</mo><mi>δ</mi><mo>)</mo></math>-LRCs with disjoint local repair groups using the first and second moment methods. As a corollary, such a random linear <math><mo>(</mo><mi>r</mi><mo>,</mo><mi>δ</mi><mo>)</mo></math>-LRC has a high probability of attaining the GV-type bound. As an analogue to the classic GV-type bound, we present two constructions of <math><mo>(</mo><mi>r</mi><mo>,</mo><mi>δ</mi><mo>)</mo></math>-LRCs that each beats this GV-type bound. One construction is obtained from a straightforward concatenation of an outer BCH code and an inner <math><mo>[</mo><mi>r</mi><mo>+</mo><mi>δ</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>r</mi><mo>]</mo></math> MDS code. Another construction is based on the Kronecker product of two matrices. To complement our results, the case that n is large but finite is also considered. In this regime, we provide an explicit upper bound for the binary r-LRCs, which is an improvement over the one in Ma and Ge (2019). Furthermore, this bound is shown to be tight for some specific parameters.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"109 ","pages":"Article 102697"},"PeriodicalIF":1.2,"publicationDate":"2025-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144596550","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}

引用次数: 0

New constructions of permutation polynomials of the form x+γTrqq3(h(x)) over finite fields with even characteristic 偶特征有限域上形式为x+γTrqq3（h(x)）的置换多项式的新构造

IF 1.2 3区数学

Finite Fields and Their Applications Pub Date : 2025-07-09 DOI: 10.1016/j.ffa.2025.102694

Sha Jiang, Kangquan Li, Longjiang Qu

引用次数: 0

Counting irreducible polynomials with restricted coefficients 计数具有限制系数的不可约多项式

IF 1.2 3区数学

Finite Fields and Their Applications Pub Date : 2025-07-07 DOI: 10.1016/j.ffa.2025.102691

Kaimin Cheng

引用次数: 0

Expansion properties of polynomials over finite fields 有限域上多项式的展开性质