Finite Fields and Their Applications最新文献

英文中文

Hermitian-Singer functional and differential codes 埃尔米特-辛格功能码和微分码

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2026-01-16 DOI: 10.1016/j.ffa.2026.102794

Gábor Korchmáros , Federico Romaniello , Valentino Smaldore

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.

厄米曲线上的代数几何码已经成为一些论文的主题，因为它们恰好具有良好的性能和大的自同构群。本文研究了由厄米曲线的辛格周期引起的问题。

引用次数: 0

De Bruijn tori without zeros: a field-theoretic perspective 没有零的德布鲁因托里：场理论视角

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2026-01-15 DOI: 10.1016/j.ffa.2026.102790

Ming Hsuan Kang, Yu Hsuan Hsieh

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.

We characterize sampling patterns as subsets whose associated field elements form an

F_{p}

-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.

我们给出了有限域上基于迹的德布鲁因托里的代数构造，重点讨论了忽略全零模式的非零变体。该构造使用两个相乘独立的生成器在环面网格上排列非零场元素，其值通过应用固定的线性映射（通常是场迹）获得。我们将采样模式描述为子集，其相关的域元素形成一个fp基，并表明列结构对应于子域上由不可约多项式决定的De Bruijn序列的循环移位。基于乘法转换的递归更新规则实现了高效的计算。

引用次数: 0

Quasi-optimum distance flag codes 准最佳距离标志码

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2026-01-15 DOI: 10.1016/j.ffa.2026.102799

Clementa Alonso-González, Miguel Ángel Navarro-Pérez

A flag is a sequence of nested subspaces of a given ambient space

F_{q}^{n}

over a finite field

F_{q}

. In network coding, a flag code is a set of flags, all of them with the same sequence of dimensions, the type vector. In this paper, we investigate quasi-optimum distance flag codes, 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 partial spreads and sunflowers. 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.

标志是给定环境空间Fqn在有限域Fq上的嵌套子空间序列。在网络编码中，标志码是一组标志，它们都具有相同的维数序列，即类型向量。在本文中，我们研究了准最优距离标志码，即那些达到次优可能距离值的标志码。我们对它们进行了表征，并给出了它们的基数的上界。此外，我们提出了一个系统的结构，为每一个选择的类型向量使用部分蔓延和向日葵。对于最小距离较低的标志码，我们调整了之前的结构，并对其表征提供了一些结果，特别是在第三最佳可能距离值的情况下。

引用次数: 0

Permutation polynomials of the form (xq−x+δ)i(q−1)+1+L(x) over Fq2 形式为（xq−x+δ）i(q−1)+1+L(x) / Fq2的置换多项式

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2026-01-14 DOI: 10.1016/j.ffa.2026.102797

Rohit Gupta , Amritanshu Rai

Let q be a power of a prime number and let

F_{q}

be the finite field with q elements. Let

δ \in F_{q^{2}}

be arbitrary. In this paper, we give a relationship between the permutation property of polynomials over

F_{q^{2}}

of the forms

g (x^{q} - x + δ) + c x^{q} + d x

and

g {(x)}^{q} - g (x) + c x^{q} + d x

where

c, d \in F_{q}^{⁎}

g (x) \in F_{q^{2}} [x]

. Further, we find the necessary and sufficient conditions on the coefficients c and d such that polynomials of the forms

{(x^{q} - x + δ)}^{i (q - 1) + 1} + c x

and

{(x^{q} - x + δ)}^{i (q - 1) + 1} + c x^{q} + d x

permute

F_{q^{2}}

. Moreover, some results of this article supersede certain results in the related literature.

设q是质数的幂，设Fq是有q个元素的有限域。设δ∈Fq2是任意的。本文给出了形式为g(xq−x+δ)+cxq+dx和g(x)q−g(x)+cxq+dx的多项式在Fq2上的置换性质之间的关系，其中c，d∈Fq, g(x)∈Fq2[x]。进一步，我们找到了系数c和d的充要条件，使得多项式的形式为（xq−x+δ）i(q−x+δ) +1+cx和（xq−x+δ）i(q−1)+1+cxq+dx可以置换Fq2。此外，本文的一些结果取代了相关文献中的某些结果。

引用次数: 0

Function-correcting codes with homogeneous distance 齐次距离函数校正码

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2026-01-13 DOI: 10.1016/j.ffa.2026.102791

Huiying Liu, Hongwei Liu

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

Z_{4}

, 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 D-homogeneous distance codes. We use D-homogeneous distance codes to characterize connections between the optimal redundancy of FCCHDs and lengths of these codes for some matrices D. 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.

功能校正码的设计是为了在保护信息的功能值不受错误影响时减少代码的冗余。作为Z4上的Hamming权和Lee权的推广，齐次权用于有限环上的码。本文引入了用FCCHDs表示的具有齐次距离的功能纠错码，它扩展了具有汉明距离的功能纠错码。首先定义d齐次距离码。我们用d -齐次距离码来描述一些矩阵d的fcchd的最优冗余度和这些码的长度之间的联系，通过这些联系，我们得到了一些函数的fcchd的最优冗余度的几个界。此外，我们还构造了齐次权函数和齐次权分布函数的FCCHDs。特别地，本文构造的一些码的冗余达到了最优冗余界。

引用次数: 0

Twisted group algebra of dihedral groups over finite fields 有限域上二面体群的扭曲群代数

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2026-01-13 DOI: 10.1016/j.ffa.2026.102792

André Duarte

Let

F_{q}

be a finite field and

D_{n}

is the dihedral group of order 2n. We present formulas for a complete set of 2-cocycles of

D_{n}

over

F_{q}

and compute the primitive central idempotents of

F_{q}^{α} D_{n}

. We conclude by describing the Wedderburn decomposition of

F_{q}^{α} D_{n}

and the irreducible projective representations of

D_{n}

over

F_{q}

设Fq是一个有限域，Dn是2n阶的二面体群。我们给出了Dn / Fq的2环完备集的公式，并计算了FqαDn的原始中心幂等。最后，我们描述了FqαDn的Wedderburn分解和Dn / Fq的不可约投影表示。

引用次数: 0

Construction of several infinite families of linear codes with new parameters: Hamming weight enumerators and hull dimensions 具有新参数：汉明权重枚举数和船体尺寸的若干无限族线性码的构造

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2026-01-13 DOI: 10.1016/j.ffa.2025.102789

Lavanya G, Anuradha Sharma

<div><div>Let q be a prime power, and let m, v, t be integers satisfying <math><mn>2</mn><mo>≤</mo><mi>t</mi><mo><</mo><mi>v</mi><mo>≤</mo><mi>m</mi></math> and <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>, where <math><mo>(</mo><mtable><mtr><mtd><mo>⋅</mo></mtd></mtr><mtr><mtd><mo>⋅</mo></mtd></mtr></mtable><mo>)</mo></math> denotes the binomial coefficient. Let X be a subset of <math><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>m</mi><mo>}</mo></math> with <math><mo>|</mo><mi>X</mi><mo>|</mo><mo>=</mo><mi>v</mi></math>. In this paper, we consider the set <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>, where <math><mtext>supp</mtext><mo>(</mo><mo>⋅</mo><mo>)</mo></math> denotes the support of a vector and <math><mi>w</mi><mo>(</mo><mo>⋅</mo><mo>)</mo></math> denotes the Hamming weight function. We first observe that the set Δ is a simplicial complex of <math><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>m</mi></mrow></msubsup></math> with support <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> consisting of all distinct subsets of X with cardinality t. Note that <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>, <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> for <math><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>b</mi></math>, and the pair <math><mo>(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo>)</mo></math> forms a trivial Steiner system. In this paper, we study linear codes over <math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math> with defining sets <math><msup><mrow><mi>Δ</mi></mrow><mrow><mi>c</mi></mrow></msup><mo>=</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</

设q为质数幂，设m、v、t为满足2≤t<；v≤m且qm−t>；(vt)≥3的整数，其中（⋅⋅）为二项式系数。设X是{1,2，…，m}的子集，其中|X|=v。本文考虑集合Δ={u∈Fqm:supp(u)≤X, w(u)≤t}，其中supp（⋅）表示向量的支持度，w（⋅）表示Hamming权函数。我们首先观察到集合Δ是支持a ={A1,A2，…，Ab}的Fqm的简单复形，由基数为t的X的所有不同子集组成。注意b=(vt)≥3,Ai∈(∈1≤j(≠i)≤bAj)=∅，对于1≤i≤b，并且对（X, a）形成一个平凡的Steiner系统。在本文中，我们研究了具有定义集Δc=Fqm∈Δ和Δ =Δ∈{0}的Fq上的线性码。我们还研究了Fq上的射影码，定义了集合和，其中和分别是Δc和Δ的极大子集，它们的向量生成Fqm在Fq上的不同的一维子空间。我们显式地确定了这些码的参数和Hamming权枚举数，并推导了具有定义集Δc和最小的码的充分条件。作为应用，我们得到了几个无限族的少权投影码、距离最优码、几乎距离最优码、Griesmer码、近Griesmer码和极小码。我们还确定了它们的双码参数。当m=v时，我们证明了具有定义集的Fq上的射影码是最优可扩展的，从而提供了一种构造Fq上无限类最优可扩展码的方法。此外，我们研究了Fq上具有定义集Δc和Δ 的线性码的壳，并证明了这些码对于q>；3是自正交的。对于q∈{2,3}，我们显式地确定了这些代码的壳体尺寸。我们也得到了无限类的二、三元自正交码、LCD码和一维壳的线性码。

引用次数: 0

A new family of maximum linear symmetric rank-distance codes 一类新的最大线性对称秩距码

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2026-01-02 DOI: 10.1016/j.ffa.2025.102787

Wei Tang , Yue Zhou

Let

S_{n} (q)

denote the set of symmetric bilinear forms over an n-dimensional

F_{q}

-vector space. A subset

C

S_{n} (q)

is called a d-code if the rank of

A - B

is larger than or equal to d for any distinct A and B in

C

. If

C

is further closed under matrix addition, then

| C |

is sharply upper bounded by

q^{n (n - d + 2) / 2}

n - d

is even and

q^{(n + 1) (n - d + 1) / 2}

n - d

is odd. Additive codes meeting these upper bounds are called maximum. There are very few known constructions of them. In this paper, we obtain a new family of maximum

F_{q}

-linear

(n - 2)

-codes in

S_{n} (q)

for

n = 6, 8

and 10 which are not equivalent to any known constructions. Furthermore, we completely determine the equivalence between distinct members in this new family.

设Sn(q)表示n维fq向量空间上对称双线性形式的集合。如果对于C中任意不同的A和B， A−B的秩大于或等于d，则Sn(q)的子集C称为d码。如果C在矩阵加法下进一步闭合，则|C|的上界是qn(n−d+2)/2，如果n−d是偶数，则q(n+1)(n−d+1)/2，如果n−d是奇数。满足这些上界的加性码称为最大值。它们的已知构造很少。本文得到了Sn(q)中n=6、8和10的最大fq -线性（n−2）码族，它们不等价于任何已知结构。此外，我们完全确定了这个新家族中不同成员之间的等价性。

引用次数: 0

One-weight codes in the sum-rank metric 一个权重在和秩度量中编码

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-12-30 DOI: 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.

单权码是指所有非零码字具有相同权值的一种高度结构化的线性码，它与有限几何结构有着密切的联系。虽然它们的分类在Hamming和rank度量中得到了很好的理解——它们等价于单纯型代码的（直接和）——但和秩度量呈现了一个复杂得多的景观。在这项工作中，我们探讨了一权和秩度量码的几何，重点是三个不同的类别。首先，我们引入并分类了常数秩表和秩度量码，其中每个非零码字具有相同的秩元组，扩展了秩度量设置的结果。接下来，我们研究更一般的常数秩-配置码，其中，直到重新排序，每个非零码字具有相同的秩元组。虽然一个完整的分类仍然难以捉摸，我们提出了第一个例子和部分结构的结果为这类。最后，我们考虑单权码也是MSRD（最大和秩距离）码。对于第2维，构造是由投影线上分散的线性集的分割产生的。对于三维空间，我们将它们的存在性与射影平面上特殊的2重块集的存在性联系起来，得到了新的界和某些域上的不存在性结果。

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引用次数: 0

Infinite families of non-simple subspace 2- and 3-designs with block dimension 4 块维为4的非简单子空间2-和3-设计的无限族

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-12-30 DOI: 10.1016/j.ffa.2025.102786

Xiaoran Wang, Junling Zhou

This paper concentrates on constructing infinite families of non-simple subspace 2-designs and 3-designs with block dimension 4. We investigate in detail the structure of the

GL (m, q^{l})

-incidence matrix between 2-subspaces and 4-subspaces of

GF {(q)}^{m l}

with

m, l \geq 3

. Employing the incidence matrix, we establish two recursive constructions for 2-

{(m l, 4, λ)}_{q}

designs, which are based on a 2-

{(l, 4, λ)}_{q}

design and a 2-

{(l, 3, μ)}_{q}

design, respectively. Several new infinite classes of simple q-analogs of group divisible designs (q-GDDs) with block dimension 4 are also produced. Making use of the recursive constructions and new q-GDDs, plenty of new infinite series of non-simple subspace 2-designs with block dimension 4 are constructed. We also study the

GL (m, q^{l})

-incidence matrix between 3-subspaces and 4-subspaces. From this, a recursive construction and a new infinite family of non-simple 3-

{(m l, 4, λ)}_{q}

designs are produced as well.

研究了块维为4的非简单子空间2-设计和3-设计无穷族的构造。研究了GF(q)ml的2-子空间和4-子空间间GL(m,ql)-关联矩阵的结构，其中m，l≥3。利用关联矩阵，分别基于2-(1,4,λ)q设计和2-(1,3,μ)q设计，建立了2-(ml,4,λ)q设计的递归结构。并给出了块维数为4的群可分设计（q- gdd）的几个新的无限类简单q-类似物。利用递归构造和新的q- gdd，构造了大量新的块维数为4的非简单子空间2-设计无穷级数。我们还研究了3子空间和4子空间之间的GL(m,ql)-关联矩阵。由此，得到了一个递归结构和一个新的非简单3-(ml,4,λ)q设计无穷族。

{"title":"Infinite families of non-simple subspace 2- and 3-designs with block dimension 4","authors":"Xiaoran Wang, Junling Zhou","doi":"10.1016/j.ffa.2025.102786","DOIUrl":"10.1016/j.ffa.2025.102786","url":null,"abstract":"<div><div>This paper concentrates on constructing infinite families of non-simple subspace 2-designs and 3-designs with block dimension 4. We investigate in detail the structure of the <math><mrow><mi>GL</mi></mrow><mo>(</mo><mi>m</mi><mo>,</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>l</mi></mrow></msup><mo>)</mo></math>-incidence matrix between 2-subspaces and 4-subspaces of <math><mi>GF</mi><mspace></mspace><msup><mrow><mo>(</mo><mi>q</mi><mo>)</mo></mrow><mrow><mi>m</mi><mi>l</mi></mrow></msup></math> with <math><mi>m</mi><mo>,</mo><mi>l</mi><mo>≥</mo><mn>3</mn></math>. Employing the incidence matrix, we establish two recursive constructions for 2-<math><msub><mrow><mo>(</mo><mi>m</mi><mi>l</mi><mo>,</mo><mn>4</mn><mo>,</mo><mi>λ</mi><mo>)</mo></mrow><mrow><mi>q</mi></mrow></msub></math> designs, which are based on a 2-<math><msub><mrow><mo>(</mo><mi>l</mi><mo>,</mo><mn>4</mn><mo>,</mo><mi>λ</mi><mo>)</mo></mrow><mrow><mi>q</mi></mrow></msub></math> design and a 2-<math><msub><mrow><mo>(</mo><mi>l</mi><mo>,</mo><mn>3</mn><mo>,</mo><mi>μ</mi><mo>)</mo></mrow><mrow><mi>q</mi></mrow></msub></math> design, respectively. Several new infinite classes of simple q-analogs of group divisible designs (q-GDDs) with block dimension 4 are also produced. Making use of the recursive constructions and new q-GDDs, plenty of new infinite series of non-simple subspace 2-designs with block dimension 4 are constructed. We also study the <math><mi>GL</mi><mo>(</mo><mi>m</mi><mo>,</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>l</mi></mrow></msup><mo>)</mo></math>-incidence matrix between 3-subspaces and 4-subspaces. From this, a recursive construction and a new infinite family of non-simple 3-<math><msub><mrow><mo>(</mo><mi>m</mi><mi>l</mi><mo>,</mo><mn>4</mn><mo>,</mo><mi>λ</mi><mo>)</mo></mrow><mrow><mi>q</mi></mrow></msub></math> designs are produced as well.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"111 ","pages":"Article 102786"},"PeriodicalIF":1.2,"publicationDate":"2025-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145883539","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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