Finite Fields and Their Applications最新文献

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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 : 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。

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引用次数: 0

Linear codes arising from the point-hyperplane geometry-Part I: The Segre embedding 由点超平面几何产生的线性码。第1部分：分段嵌入

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-12-02 DOI: 10.1016/j.ffa.2025.102766

I. Cardinali , L. Giuzzi

Let V be a vector space over the finite field

F_{q}

with q elements and Λ be the image of the Segre geometry

PG (V) \otimes PG (V^{⁎})

PG (V \otimes V^{⁎})

under the Segre map. Consider the subvariety

Λ_{1}

of Λ represented by the pure tensors

x \otimes ξ

with

x \in V

and

ξ \in V^{⁎}

such that

ξ (x) = 0

. Regarding

Λ_{1}

as a projective system of

PG (V \otimes V^{⁎})

, we study the linear code

C (Λ_{1})

arising from it. We show that

C (Λ_{1})

is a minimal code and we determine its basic parameters, its full weight list and its linear automorphism group. We also give a geometrical characterization of its minimum and second lowest weight codewords as well as of some of the words of maximum weight.

设V为具有q个元素的有限域Fq上的向量空间，Λ为Segre几何图形PG(V)⊗PG(V)在PG（V V）中在Segre映射下的像。考虑由x∈V和ξ∈V的纯张量x⊗ξ表示的Λ的子变种Λ1，使得ξ(x)=0。将Λ1看作PG（V⊗V）的一个射影系统，研究了由此产生的线性代码C（Λ1）。我们证明了C（Λ1）是一个最小码，并确定了它的基本参数、它的全权表和它的线性自同构群。我们还给出了它的最小码字和次最小码字以及一些最大码字的几何特征。

引用次数: 0

Permutation polynomials and involutions over the finite field F22m 有限域F22m上的置换多项式与对折

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-12-01 DOI: 10.1016/j.ffa.2025.102760

Liqin Qian , Minjia Shi , Xiwang Cao

In this paper, we propose several classes of permutation polynomials of the form

\sum_{j = 1}^{t} {({Tr}_{m}^{2 m} {(x)}^{k_{j}} + δ_{j})}^{s_{j}} + L (x)

over

F_{2^{2 m}}

, where

L (x) = a {Tr}_{m}^{2 m} (x) + b x

a \in F_{2^{2 m}}

and

b \in F_{2^{m}}^{⁎}

. The permutation behavior of the proposed polynomials is investigated by the AGW criterion and determination of the number of solutions to certain equations over

F_{2^{2 m}}

. Based on an effective method proposed by Mesnager (2014), we construct several classes of involutions and further obtain some self-dual bent functions by employing three permutations of

F_{2^{2 m}}

satisfying an algebraic property

(A_{2 m})

. Finally, it is worth pointing out that there exist examples of bent functions we obtained which do not belong to

M M^{#}

本文提出了几类形式为∑j=1t(Trm2m(x)kj+δj)sj+L(x) / F22m的置换多项式，其中L(x)=aTrm2m(x)+bx, a∈F22m, b∈F2m。利用AGW准则和确定F22m上某些方程的解的个数，研究了所提出多项式的置换行为。在Mesnager（2014）提出的有效方法的基础上，利用满足代数性质（A2m）的F22m的三个排列，构造了几类对合，并进一步得到了一些自对偶弯曲函数。最后，值得指出的是，我们得到的弯曲函数也存在不属于mm#的例子。

引用次数: 0

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 : 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

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 : 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

On the PGL2(q)-orbits of lines of PG(3,q) and binary quartic forms in characteristic three 特征三中PG（3,q）和二元四次型线的PGL2(q)轨道

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-11-26 DOI: 10.1016/j.ffa.2025.102763

Krishna Kaipa , Puspendu Pradhan

We consider the problem of classifying the lines of the projective 3-space

P G (3, q)

over a finite field

F_{q}

into orbits of the group

P G L_{2} (q)

of linear symmetries of the twisted cubic C. The problem has been solved in literature in characteristic different from 3, and in this work, we solve the problem in characteristic 3. We reduce this problem to another problem, which is the classification of binary quartic forms into

P G L_{2} (q)

-orbits. We first solve the latter problem and use to solve the former problem. We also obtain the point-line and the line-plane incidence structures of the point, line, and plane orbits.

考虑有限域Fq上的射影3-空间PG（3,q）的线划分为扭曲三次c的线性对称群PGL2(q)的轨道的问题。这个问题已经在不同于特征3的文献中得到了解决，在本文中，我们解决了特征3中的问题。我们将这个问题简化为另一个问题，即二元四次形式在PGL2(q)轨道中的分类问题。我们先解决后一个问题，再解决前一个问题。我们还得到了点、线、面轨道的点-线和线-面入射结构。

引用次数: 0

Explicit representatives and sizes of cyclotomic cosets and their application to cyclic codes over finite fields 有限域上环切集的显式表示和大小及其在循环码中的应用

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-11-25 DOI: 10.1016/j.ffa.2025.102761

Li Zhu , Jinle Liu , Hongfeng Wu

Cyclotomic coset is a classical notion in the theory of finite fields which has wide applications in various computation problems. Let q be a prime power, and n be a positive integer coprime to q. In this paper we determine explicitly the representatives and the sizes of all q-cyclotomic cosets modulo n in the general settings. We introduce the definition of 2-adic cyclotomic system, which is a profinite space consists of certain compatible sequences of cyclotomic cosets. A precise characterization of the structure of the 2-adic cyclotomic system is given, which reveals the general formula for representatives of cyclotomic cosets. With the representatives and the sizes of q-cyclotomic cosets modulo n, we improve the formulas for the factorizations of

X^{n} - 1

and of

Φ_{n} (X)

over

F_{q}

given in [4]. As a consequence, we classify the cyclic codes over finite fields via giving their generator polynomials. Moreover, the self-dual cyclic codes are determined and enumerated.

分环协集是有限域理论中的一个经典概念，在各种计算问题中有着广泛的应用。设q为素数幂，n为q的正整数的协素数。在一般情况下，我们明确地确定了以n为模的所有q-环形协集的表示和大小。引入了二进切环系统的定义，该系统是由若干切环协集相容序列组成的无限空间。给出了二进分环系统结构的一个精确表征，并由此得到了分环系统的共集表示的一般公式。利用以n为模的q-环形集的表示和大小，改进了[4]中给出的Xn−1和Φn(X) / Fq的分解公式。因此，我们通过给出循环码的生成多项式对有限域上的循环码进行分类。此外，还确定并列举了自对偶循环码。

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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 : 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

On the Hermitian Veronesean 在厄米特的维罗内西亚

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-11-21 DOI: 10.1016/j.ffa.2025.102762

John Bamberg , Geertrui Van de Voorde

The Hermitian Veronesean in

PG (3, q^{2})

, given by

V : = {(1, x, x^{q}, x^{q + 1}) : x \in F_{q^{2}}} \cup {(0, 0, 0, 1)}

, is a well-studied rational curve, and forms a special set of the Hermitian surface

H (3, q^{2})

. In this paper, we give two local characterisations of the Hermitian Veronesean, based on sublines and triples of points in perspective.

由V:={(1,x,xq,xq+1):x∈Fq2}∪{(0,0,0,1)}给出的PG（3,q2）中的厄米维罗内塞曲线是一个被充分研究的有理曲线，它构成了厄米曲面H（3,q2）的一个特殊集合。在本文中，我们给出了基于透视点的子线和三元组的厄米维罗内式的两个局部特征。

引用次数: 0

A provably quasi-polynomial algorithm for the discrete logarithm problem in finite fields of small characteristic 小特征有限域中离散对数问题的可证明拟多项式算法

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-11-20 DOI: 10.1016/j.ffa.2025.102753

Guido Lido

We describe a provably quasi-polynomial algorithm to compute discrete logarithms in the multiplicative groups of finite fields of small characteristic, that is finite fields whose characteristic is logarithmic in the order. We partially follow the heuristically quasi-polynomial algorithm presented by Barbulescu, Gaudry, Joux and Thomé. The main difference is to use a presentation of the finite field based on elliptic curves: the abundance of elliptic curves ensures the existence of such a presentation.

描述了一种计算小特征有限域（即特征为对数阶的有限域）乘积群离散对数的可证明拟多项式算法。我们部分遵循Barbulescu， Gaudry， Joux和thom提出的启发式拟多项式算法。主要区别在于使用基于椭圆曲线的有限域表示：椭圆曲线的丰度保证了这种表示的存在。

引用次数: 0

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