Finite Fields and Their Applications最新文献

英文中文

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。

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引用次数: 0

Quantum (r,δ)-locally recoverable codes 量子(r,δ)-局部可恢复的代码

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-12-30 DOI: 10.1016/j.ffa.2025.102785

Carlos Galindo , Fernando Hernando , Helena Martín-Cruz , Ryutaroh Matsumoto

Classical

(r, δ)

-locally recoverable codes are designed for avoiding loss of information in large scale distributed and cloud storage systems. We introduce the quantum counterpart of those codes by defining quantum

(r, δ)

-locally recoverable codes which are quantum error-correcting codes capable of correcting

δ - 1

qudit erasures from sets of at most

r + δ - 1

qudits.

We give a necessary and sufficient condition for a quantum stabilizer code

Q (C)

to be

(r, δ)

-locally recoverable. Our condition depends only on the puncturing and shortening at suitable sets of both the symplectic self-orthogonal code C used for constructing

Q (C)

and its symplectic dual

C^{⊥_{s}}

. When

Q (C)

comes from a Hermitian or Euclidean dual-containing code, and under an extra condition, we show that there is an equivalence between the classical and quantum concepts of

(r, δ)

-local recoverability. A Singleton-like bound is stated in this case and examples attaining the bound are given.

经典的（r,δ）本地可恢复代码是为避免大规模分布式和云存储系统中的信息丢失而设计的。我们通过定义量子(r,δ)-局部可恢复码来引入这些码的量子对偶，这些码是量子纠错码，能够从最多r+δ−1个量子比特的集合中纠正δ−1个量子比特的擦除。给出了量子稳定码Q(C)是（r,δ）局部可恢复的充分必要条件。我们的条件只依赖于用于构造Q(C)的辛自正交码C及其辛对偶C⊥在合适集合上的穿刺和缩短。当Q(C)来自厄米码或欧几里得双包含码时，在一个额外的条件下，我们证明了（r,δ）局部可恢复性的经典概念与量子概念之间存在等价性。在这种情况下，给出了一个类单例边界，并给出了实现该边界的例子。

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引用次数: 0

Kneser's theorem for codes and ℓ-divisible set families 码和可分集合族的Kneser定理

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-12-17 DOI: 10.1016/j.ffa.2025.102783

Chenying Lin , Gilles Zémor

A k-wise ℓ-divisible set family is a collection

F

of subsets of

{1, \dots, n}

such that any intersection of k sets in

F

has cardinality divisible by ℓ. If

k = ℓ = 2

, it is well-known that

| F | \leq 2^{⌊ n / 2 ⌋}

. We generalise this by proving that

| F | \leq 2^{⌊ n / p ⌋}

k = ℓ = p

, for any prime number p.

For arbitrary values of ℓ, we prove that

4 ℓ^{2}

-wise ℓ-divisible set families

F

satisfy

| F | \leq 2^{⌊ n / ℓ ⌋}

and that the only families achieving the upper bound are atomic, meaning that they consist of all the unions of disjoint subsets of size ℓ. This improves upon a recent result by Gishboliner, Sudakov and Timon, that arrived at the same conclusion for k-wise ℓ-divisible families, with values of k that behave exponentially in ℓ.

Our techniques rely heavily upon a coding-theory analogue of Kneser's Theorem from additive combinatorics.

一个向k可整除的集合族是{1，…，n}的子集的集合F，使得F中k个集合的任何交集都具有可被r整除的基数。若k= n =2，则已知|F|≤2⌊n/2⌋。我们通过证明|F|≤2⌊n/p⌋，如果k= r =p，对于任意素数p，我们证明了4个2 ~ 2可分集合族F满足|F|≤2⌊n/p⌋，并且唯一达到上限的族是原子族，这意味着它们由大小为r的不相交子集的所有并组成。这改进了Gishboliner， Sudakov和Timon最近的一个结果，他们对k-可分族得出了相同的结论，其中k的值在r中表现为指数。我们的技术在很大程度上依赖于可加组合学中克尼泽定理的编码理论类比。

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引用次数: 0

The Rado multiplicity problem in vector spaces over finite fields 有限域上向量空间的Rado多重性问题

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-12-17 DOI: 10.1016/j.ffa.2025.102782

Juanjo Rué , Christoph Spiegel

We study an analogue of the Ramsey multiplicity problem for additive structures, in particular establishing the minimum number of monochromatic 3-APs in 3-colorings of

F_{3}^{n}

as well as obtaining the first non-trivial lower bound for the minimum number of monochromatic 4-APs in 2-colorings of

F_{5}^{n}

. The former parallels results by Cumings et al. [8] in extremal graph theory and the latter improves upon results of Saad and Wolf [42]. The lower bounds are notably obtained by extending the flag algebra calculus of Razborov [39] to additive structures in vector spaces over finite fields.

我们研究了可加性结构Ramsey多重性问题的一个类似问题，特别是建立了F3n的3-着色中单色3- ap的最小数目，以及F5n的2-着色中单色4- ap的最小数目的第一个非平凡下界。前者与Cumings et al.[8]在极值图论中的结果相似，后者改进了Saad和Wolf[8]的结果。将Razborov[39]的标志代数演算推广到有限域上向量空间的加性结构，得到了下界。

引用次数: 0

New curves of Kummer type with many rational points over finite fields 有限域上具有多有理点的Kummer型新曲线

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-12-16 DOI: 10.1016/j.ffa.2025.102780

H. Navarro, Luz A. Pérez

In this paper, we present two methods for constructing curves of Kummer type with many rational points over finite fields. The first method is based on binomials, while the second employs reciprocal polynomials. The latter is an extension of the method introduced by Gupta et al. (2023) [19] over quadratic finite fields, to non-prime finite fields. As a result, we found 63 new records and 37 new entries for the online table of curves with many points found at manYPoints.

本文给出了在有限域上构造具有多有理点的Kummer型曲线的两种方法。第一种方法是基于二项式，而第二种方法是使用互反多项式。后者是将Gupta等人（2023）[19]在二次有限域上引入的方法推广到非素数有限域。结果，我们发现了63条新记录和37个新条目，用于在线曲线表，其中在manYPoints上发现了许多点。

引用次数: 0

One class of maximal cliques in the collinearity graphs of geometries related to simplex codes 与单纯形码相关的几何共线性图中的一类极大团

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-12-15 DOI: 10.1016/j.ffa.2025.102784

Mariusz Kwiatkowski, Mark Pankov, Adam Tyc

Consider the point-line geometry

S (k, q)

whose maximal singular subspaces correspond to q-ary simplex codes of dimension k. Maximal cliques in the collinearity graph of this geometry contain no more than

n = (q^{k} - 1) / (q - 1)

elements and maximal singular subspaces of

S (k, q)

are n-cliques of this graph. If

q = 2

, then

n = 2^{k} - 1

and there is a one-to-one correspondence between

(2^{k} - 1)

-cliques of the collinearity graph and symmetric

(2^{k} - 1, 2^{k - 1}, 2^{k - 2})

-designs. For the case when

q \geq 5

we construct a class of n-cliques distinct from maximal singular subspaces. In the case when

k = 2

, some of these cliques are normal rational curves.

考虑点线几何S(k,q)，其极大奇异子空间对应于k维的q元单纯形码。该几何的共线性图中的极大团包含不超过n=(qk−1)/(q−1)个元素，并且S（k,q）的极大奇异子空间为该图的n个团。如果q=2，则n=2k−1，并且共线性图的(2k−1)-团与对称(2k−1,2k−1,2k−2)-设计之间存在一一对应关系。对于q≥5的情况，我们构造了一类不同于极大奇异子空间的n-团。在k=2的情况下，其中一些团是正态有理曲线。

引用次数: 0

On the addition of squares and cubes of units modulo n 关于以n为模的单位的平方和立方的加法

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-12-15 DOI: 10.1016/j.ffa.2025.102778

Yajing Zhou , Rongquan Feng

Let

Z_{n}

be the ring of residue classes modulo n, and let

Z_{n}^{⁎}

be the group of its units. In 2017, Mollahajiaghaei presented a formula for the number of solutions

(x_{1}, . . ., x_{k}) \in {(Z_{n}^{⁎})}^{k}

of the congruence

x_{1}^{2} + \dots + x_{k}^{2} \equiv c (\mod n)

. This paper considers the addition of squares and cubes over

Z_{n}^{⁎}

. Specifically, when n is a prime number such that

n \equiv 1 (\mod 4)

, we correct the formula given by Mollahajiaghaei.

设Zn为模n的残馀类环，设Zn为它的单元群。2017年，Mollahajiaghaei提出了解决方案数量的公式(x1，…，xk)∈(Zn _)k的同余式x12+⋯+xk2≡c（modn）。本文考虑Zn - z上的平方和立方的加法。具体地说，当n是质数使得n≡1（mod4）时，我们修正Mollahajiaghaei给出的公式。

引用次数: 0

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