Finite Fields and Their Applications最新文献

英文中文

On two conjectures of the duals of AMDS BCH codes 关于AMDS BCH码对偶的两个猜想

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2026-03-01 Epub Date: 2025-12-01 DOI: 10.1016/j.ffa.2025.102765

Haode Yan

In this paper, we study the duals of BCH codes

C_{(q, q + 1, 3, \frac{p - 1}{2})}

and

C_{(3^{s}, 3^{s} + 1, 3, 4)}

. By analyzing the number of solutions to certain equations over the multiplicative subgroup

U_{q + 1} = {x \in F_{q^{2}} : x^{q + 1} = 1}

F_{q^{2}}^{⁎}

, we determine the possible weights of codewords in

C_{(q, q + 1, 3, \frac{p - 1}{2})}^{⊥}

and

C_{(3^{s}, 3^{s} + 1, 3, 4)}^{⊥}

, respectively. The weight distributions of these two dual codes are derived by applying the Pless power moments. Our results provide affirmative solutions to recent conjectures.

本文研究了BCH码C（q,q+1,3,p−12）和C（3s,3s+1,3,4）的对偶。通过分析Fq2的乘法子群Uq+1={x∈Fq2:xq+1=1}上某些方程的解的个数，我们分别确定了C(q,q+1,3,p−12)⊥和C(3s,3s+1,3,4)⊥中码字的可能权重。应用无功矩导出了这两种双码的权值分布。我们的结果为最近的猜想提供了肯定的答案。

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 : 2026-03-01 Epub 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

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 : 2026-03-01 Epub 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

A family of optimal dual-containing and reversible linear codes over F4 F4上最优的双含可逆线性码族

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2026-03-01 Epub Date: 2025-12-11 DOI: 10.1016/j.ffa.2025.102777

Sujata Bansal, Pramod Kumar Kewat

In this paper, we construct a family of optimal linear codes over

F_{4}

with parameters

[2^{e}, 2^{e} - e - 1, 4]

, where e is a positive integer and

e \geq 2

. We determine the duals of these codes and establish that for

e \geq 3

, these codes are dual-containing. This property makes them suitable for the construction of CSS quantum error-correcting codes. Furthermore, we calculate the weight distribution of the duals of these codes and show that the duals are 3-weight codes. We derive the weight enumerator of these codes using the MacWilliams identities. Additionally, we establish that these codes are reversible for all

e \geq 2

. This ensures the symmetry in the code structure and facilitates them for the possible applications in DNA computing and bidirectional communication systems. The optimality, duality, and reversibility of this family of codes highlight the potential of these codes for various practical and theoretical applications in the error correction.

本文构造了F4上具有参数[2e，2e−e−1,4]的最优线性码族，其中e为正整数且e≥2。我们确定了这些码的对偶，并建立了当e≥3时，这些码是双包含的。这种特性使它们适合于构建CSS量子纠错码。进一步，我们计算了这些码的对偶码的权值分布，并证明了对偶码是3权码。我们利用MacWilliams恒等式导出了这些码的权重枚举数。此外，我们还证明了这些编码对于所有e≥2都是可逆的。这保证了编码结构的对称性，并为它们在DNA计算和双向通信系统中的可能应用提供了便利。这组码的最优性、对偶性和可逆性突出了这些码在纠错中的各种实际和理论应用的潜力。

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引用次数: 0

Cycles and cuts in supersingular L-isogeny graphs 超奇异l -等构图中的环与切

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2026-03-01 Epub Date: 2025-12-11 DOI: 10.1016/j.ffa.2025.102768

Sarah Arpin , Ross Bowden , James Clements , Wissam Ghantous , Jason T. LeGrow , Krystal Maughan

Supersingular elliptic curve isogeny graphs underlie isogeny-based cryptography. For isogenies of a single prime degree ℓ, their structure has been investigated graph-theoretically. We generalise the notion of ℓ-isogeny graphs to L-isogeny graphs (studied in the prime field case by Delfs and Galbraith), where L is a set of small primes dictating the allowed isogeny degrees in the graph. We analyse the graph-theoretic structure of L-isogeny graphs. Our approaches may be put into two categories: cycles and graph cuts.

On the topic of cycles, we provide: a count for the number of cycles in the L-isogeny graph with cyclic kernels using traces of Brandt matrices; an efficiently computable estimate based on this approach; and a third ideal-theoretic count for a certain subclass of L-isogeny cycles. We provide code to compute each of these three counts.

On the topic of graph cuts, we compare several algorithms to compute graph cuts which minimise a measure called the edge expansion, outlining a cryptographic motivation for doing so. Our results show that a greedy neighbour algorithm out-performs standard spectral algorithms for computing optimal graph cuts. We provide code and study explicit examples.

Furthermore, we describe several directions of active and future research.

超奇异椭圆曲线等构图是等构密码学的基础。对于单素数阶的同胚，我们用图理论研究了它们的结构。我们将L-等构图的概念推广到L-等构图（由Delfs和Galbraith在素场情况下研究），其中L是一组小素数，表示图中允许的等构度。我们分析了l -等构图的图论结构。我们的方法可以分为两类：循环和图切。在循环的主题上，我们提供了：使用Brandt矩阵的迹来计算具有循环核的l -等同图中的循环数；基于该方法的高效可计算估计；以及l-等同系环的某个子类的第三个理想理论计数。我们提供了计算这三种计数的代码。关于图割的主题，我们比较了几种算法来计算图割，这些算法最小化了称为边缘扩展的度量，概述了这样做的密码学动机。我们的结果表明，贪婪邻居算法在计算最优图切割方面优于标准谱算法。我们提供代码并研究显式示例。此外，我们描述了几个活跃的和未来的研究方向。

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引用次数: 0

Multiplicative character sums over two classes of subsets of quadratic extensions of finite fields 有限域的二次扩展的两类子集上的乘法字符和

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2026-03-01 Epub Date: 2025-12-03 DOI: 10.1016/j.ffa.2025.102767

Kaimin Cheng , Arne Winterhof

Let q be a prime power and r a positive even integer. Let

F_{q}

be the finite field with q elements and

F_{q^{r}}

be its extension field of degree r. Let χ be a nontrivial multiplicative character of

F_{q^{r}}

and

f (X)

a polynomial over

F_{q^{r}}

with exactly one simple root in

F_{q^{r}}

. In this paper, we improve estimates for character sums

\sum_{g \in G} χ (f (g))

, where

G

is either a subset of

F_{q^{r}}

of sparse elements, with respect to some fixed basis of

F_{q^{r}}

which contains a basis of

F_{q^{r / 2}}

, or a subset avoiding affine hyperplanes in general position. While such sums have been previously studied, our approach yields sharper bounds by reducing them to sums over the subfield

F_{q^{r / 2}}

rather than sums over general linear spaces. These estimates can be used to prove the existence of primitive elements in

G

in the standard way.

设q为质数幂，r为正偶数。设Fq是有q个元素的有限域，Fqr是它的r次扩展域。设χ是Fqr和f(X)的非平凡乘性，f(X)是Fqr上的一个多项式，在Fqr上只有一个单根。在本文中，我们改进了特征和∑g∈Gχ(f(g))的估计，其中g是稀疏元素的Fqr的子集，关于Fqr的某个固定基，其中包含Fqr/2的基，或者是在一般位置上避免仿射超平面的子集。虽然以前已经研究过这样的和，但我们的方法通过将它们简化为子域Fqr/2上的和而不是一般线性空间上的和而产生了更清晰的界限。这些估计可以用标准的方法证明G中原元的存在性。

{"title":"Multiplicative character sums over two classes of subsets of quadratic extensions of finite fields","authors":"Kaimin Cheng , Arne Winterhof","doi":"10.1016/j.ffa.2025.102767","DOIUrl":"10.1016/j.ffa.2025.102767","url":null,"abstract":"<div><div>Let <em>q</em> be a prime power and <em>r</em> a positive even integer. 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 and <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>r</mi></mrow></msup></mrow></msub></math></span> be its extension field of degree <em>r</em>. Let <em>χ</em> be a nontrivial multiplicative character of <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>r</mi></mrow></msup></mrow></msub></math></span> and <span><math><mi>f</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> a polynomial over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>r</mi></mrow></msup></mrow></msub></math></span> with exactly one simple root in <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>r</mi></mrow></msup></mrow></msub></math></span>. In this paper, we improve estimates for character sums <span><math><munder><mo>∑</mo><mrow><mi>g</mi><mo>∈</mo><mi>G</mi></mrow></munder><mi>χ</mi><mo>(</mo><mi>f</mi><mo>(</mo><mi>g</mi><mo>)</mo><mo>)</mo></math></span>, where <span><math><mi>G</mi></math></span> is either a subset of <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>r</mi></mrow></msup></mrow></msub></math></span> of sparse elements, with respect to some fixed basis of <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>r</mi></mrow></msup></mrow></msub></math></span> which contains a basis of <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>r</mi><mo>/</mo><mn>2</mn></mrow></msup></mrow></msub></math></span>, or a subset avoiding affine hyperplanes in general position. While such sums have been previously studied, our approach yields sharper bounds by reducing them to sums over the subfield <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>r</mi><mo>/</mo><mn>2</mn></mrow></msup></mrow></msub></math></span> rather than sums over general linear spaces. These estimates can be used to prove the existence of primitive elements in <span><math><mi>G</mi></math></span> in the standard way.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"111 ","pages":"Article 102767"},"PeriodicalIF":1.2,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145684471","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

Quantum MDS codes induced by the projective linear transformation 投影线性变换诱导的量子MDS码

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2026-03-01 Epub Date: 2025-12-01 DOI: 10.1016/j.ffa.2025.102764

Fengwei Li, Yuting Liu, Ruiyuan Jiang

Let

F_{q}

be the finite field with q elements, where q is a power of an odd prime p. In this paper, we provide a method to construct Hermitian self-orthogonal generalized Reed-Solomon (GRS) codes and extended GRS codes, which their support sets are roots of polynomials from affine and projective linear transformation over

F_{q^{2}}

. Moreover, we construct three classes of quantum maximum distance separable (MDS) codes with minimum distances

> \frac{q}{2} + 1

. Some of these quantum MDS codes have not been obtained before, and in some cases, have larger minimum distances and higher efficiency than the well-known quantum MDS codes.

设Fq是有q个元素的有限域，其中q是奇素数p的幂。本文给出了一种构造hermite自正交广义里德-所罗门码（GRS）和扩展GRS码的方法，它们的支持集是Fq2上仿射和射影线性变换的多项式的根。此外，我们构造了3类最小距离为>；q2+1的量子最大距离可分离码（MDS）。这些量子MDS码有些是以前没有得到过的，在某些情况下，比已知的量子MDS码具有更大的最小距离和更高的效率。

引用次数: 0

Maps preserving a fixed rank-distance on matrices over finite fields 在有限域上矩阵上保持固定秩距的映射

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2026-03-01 Epub Date: 2025-11-25 DOI: 10.1016/j.ffa.2025.102759

A.M. Maksaev, N.Y. Medved, V.V. Promyslov

Denote by

M_{m \times n}

the space of all

m \times n

matrices over a field. For a fixed

1 ⩽ k ⩽ \min (m, n)

, we investigate bijective maps

φ_{1}, φ_{2} : M_{m \times n} \to M_{m \times n}

such that

rk (A - B) = k

iff

rk (φ_{1} (A) - φ_{2} (B)) = k

, for any

A, B \in M_{m \times n}

. When

k < \min (m, n) / 2

, we not only characterize such maps on matrix spaces, but prove that such maps are equal isometries even on more general metric spaces that we call discrete-triangular. For an arbitrary k, we prove that the same characterization holds for the matrices over finite fields, except for

2 \times 2

matrices over the field of 2 elements. To do this, we use theory of association schemes, specifically the bilinear forms scheme, and investigate its eigenvalues and intersection numbers.

用Mm×n表示一个域上所有m×n矩阵的空间。对于一个固定的1≤k≤min (m,n)，我们研究了双射映射φ1，φ2:Mm×n→Mm×n，使得rk(a−B)=k且rk(φ1(a)−φ2(B))=k，对于任意a，B∈Mm×n。当k<；min (m,n)/2时，我们不仅在矩阵空间上刻画了这样的映射，而且证明了这样的映射在更一般的度量空间上是相等的等距，我们称之为离散三角空间。对于任意k，我们证明了除了2×2 2元域上的矩阵外，有限域上的矩阵也具有相同的性质。为此，我们使用关联格式理论，特别是双线性格式，并研究其特征值和交数。

引用次数: 0

Two classes of NMDS codes from Roth-Lempel codes 从Roth-Lempel码中得到两类NMDS码

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2026-03-01 Epub Date: 2025-12-12 DOI: 10.1016/j.ffa.2025.102779

Zhonghao Liang, Qunying Liao

Since near maximum distance separable (NMDS) codes have good algebraic properties and excellent error-correcting capabilities, they have been widely used in various fields such as communication systems, data storage, quantum codes, and so on. In this paper, basing on the generator matrix of Roth-Lempel codes, we present two classes of NMDS codes which generalize Han's, Zheng's and Zhou's constructions in 2023 and 2025, respectively. And we also completely determine their weight distributions.

由于近最大距离可分离码具有良好的代数性质和良好的纠错能力，在通信系统、数据存储、量子码等领域得到了广泛的应用。本文基于Roth-Lempel码的生成矩阵，提出了两类NMDS码，分别推广了2023年和2025年的Han、Zheng和Zhou的结构。我们也完全确定了它们的权重分布。

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 : 2026-03-01 Epub 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

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