Finite Fields and Their Applications最新文献

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Construction of Galois self-orthogonal MDS codes with larger dimensions 大维伽罗瓦自正交MDS码的构造

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-12-01 Epub Date: 2025-06-06 DOI: 10.1016/j.ffa.2025.102665

Ruhao Wan, Shixin Zhu

<div><div>Let <math><mi>q</mi><mo>=</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>m</mi></mrow></msup></math> be a prime power, e be an integer with <math><mn>0</mn><mo>≤</mo><mi>e</mi><mo>≤</mo><mi>m</mi><mo>−</mo><mn>1</mn></math> and <math><mi>s</mi><mo>=</mo><mi>gcd</mi><mo>⁡</mo><mo>(</mo><mi>e</mi><mo>,</mo><mi>m</mi><mo>)</mo></math>. In this paper, for a vector <math><mi>v</mi><mo>∈</mo><msup><mrow><mo>(</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup><mo>)</mo></mrow><mrow><mi>n</mi></mrow></msup></math> and a q-ary linear code <math><mi>C</mi></math>, we give some necessary and sufficient conditions for the equivalent code <math><msub><mrow><mi>Φ</mi></mrow><mrow><mi>v</mi></mrow></msub><mo>(</mo><mi>C</mi><mo>)</mo></math> of <math><mi>C</mi></math> and the extended code of <math><msub><mrow><mi>Φ</mi></mrow><mrow><mi>v</mi></mrow></msub><mo>(</mo><mi>C</mi><mo>)</mo></math> to be e-Galois self-orthogonal. We then directly obtain some necessary and sufficient conditions for (extended) generalized Reed-Solomon (GRS and EGRS) codes to be e-Galois self-orthogonal. From this we show that if <math><mi>k</mi><mo>≥</mo><mi>min</mi><mo>⁡</mo><mo>{</mo><mi>max</mi><mo>⁡</mo><mo>{</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>e</mi></mrow></msup><mo>,</mo><mo>⌈</mo><mfrac><mrow><mi>n</mi><mo>+</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>e</mi></mrow></msup></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>e</mi></mrow></msup><mo>+</mo><mn>1</mn></mrow></mfrac><mo>⌉</mo><mo>}</mo><mo>,</mo><mi>max</mi><mo>⁡</mo><mo>{</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>m</mi><mo>−</mo><mi>e</mi></mrow></msup><mo>,</mo><mo>⌈</mo><mfrac><mrow><mi>n</mi><mo>+</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>m</mi><mo>−</mo><mi>e</mi></mrow></msup></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>m</mi><mo>−</mo><mi>e</mi></mrow></msup><mo>+</mo><mn>1</mn></mrow></mfrac><mo>⌉</mo><mo>}</mo><mo>}</mo></math>, there is no <math><msub><mrow><mo>[</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>]</mo></mrow><mrow><mi>q</mi></mrow></msub></math> e-Galois self-orthogonal (extended) GRS code. Furthermore, for all possible e satisfying <math><mn>0</mn><mo>≤</mo><mi>e</mi><mo>≤</mo><mi>m</mi><mo>−</mo><mn>1</mn></math>, we classify them into three cases (1) <math><mfrac><mrow><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></math> odd and p even; (2) <math><mfrac><mrow><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></math> odd and p odd; (3) <math><mfrac><mrow><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></math> even, and construct several new classes of e-Galois self-orthogonal maximum distance separable (MDS) codes. It is worth noting that our e-Galois self-orthogonal MDS

设q=pm为质数幂，e为0≤e≤m−1的整数，s=gcd （e,m）。本文针对向量v∈(Fq)n和一个q元线性码C，给出了C的等价码Φv(C)和扩展码Φv(C)是e-伽罗瓦自正交的一些充要条件。然后直接得到了广义Reed-Solomon码（GRS和EGRS）是e-伽罗瓦自正交的一些充要条件。由此我们证明了如果k≥min (min) (max) (pe,≤n+pepe+1), (max) （pm−e,≤n+pm−epm−e+1）²}}，不存在[n，k]q e-伽罗瓦自正交（扩展）GRS码。进一步，对于所有可能满足0≤e≤m−1的e，我们将它们分为三种情况(1)ms为奇数，p为偶数；(2) ms奇数和p奇数；(3)构造了几类新的e-伽罗瓦自正交最大距离可分离码（MDS）。值得注意的是，我们的e-伽罗瓦自正交MDS码的维数可以大于⌊n+pe−1pe+1⌋，这是以前已知的码所没有的。此外，根据传播规则，我们得到了一些新的具有任意维伽罗瓦壳的MDS码。

引用次数: 0

The compositional inverses of permutation polynomials of the form ∑i=1kbi(xpm+x+δ)si−x over Fp2m 形式为∑i=1kbi(xpm+x+δ)si−x / Fp2m的排列多项式的组合逆

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-12-01 Epub Date: 2025-06-12 DOI: 10.1016/j.ffa.2025.102681

Danyao Wu , Pingzhi Yuan , Huanhuan Guan , Juan Li

In this paper, we present the compositional inverses of several classes permutation polynomials of the form

\sum_{i = 1}^{k} b_{i} {(x^{p^{m}} + x + δ)}^{s_{i}} - x

over

F_{p^{2 m}}

, where for

1 \leq i \leq k

s_{i}, m

are positive integers,

b_{i}, δ \in F_{p^{2 m}}

, and p is prime.

本文给出了形式为∑i=1kbi(xpm+x+δ)si−x / Fp2m的几类置换多项式的组合逆，其中当1≤i≤k时，si，m为正整数，bi,δ∈Fp2m， p为素数。

引用次数: 0

Permutation polynomials of finite fields of even characteristic from character sums 从字符和看偶特征有限域的置换多项式

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-12-01 Epub Date: 2025-06-30 DOI: 10.1016/j.ffa.2025.102684

Ruikai Chen , Sihem Mesnager

In this paper, we investigate permutation polynomials over the finite field

F_{q^{n}}

with

q = 2^{m}

, focusing on those in the form

Tr (A x^{q + 1}) + L (x)

, where

A \in F_{q^{n}}^{⁎}

and L is a 2-linear polynomial over

F_{q^{n}}

. By calculating certain character sums, we characterize these permutation polynomials and provide additional constructions.

本文研究了有限域Fqn上q=2m的置换多项式，重点研究了形式为Tr(Axq+1)+L(x)的置换多项式，其中A∈Fqn， L是Fqn上的2-线性多项式。通过计算某些特征和，我们描述了这些排列多项式，并提供了额外的结构。

引用次数: 0

A tighter bound on the minimum distances for an infinite family of binary BCH codes and its generalization 无限族二进制BCH码的最小距离的更紧界及其推广

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-12-01 Epub Date: 2025-05-08 DOI: 10.1016/j.ffa.2025.102628

Haodong Lu, Xuan Wang, Minjia Shi

In this paper, we improve the bound on the minimum distance for the family of binary cyclic codes proposed by Sun et al. (2024) [8]. The 3-ary analogue is also studied in this paper, which is a nice family of ternary cyclic codes that contains some best known linear codes, and this family has a better lower bound on minimum distance than that of codes proposed by Chen et al. (2023) [2].

本文改进了Sun et al.(2024)[8]提出的二进制循环码族的最小距离界。本文还研究了3元模拟，这是一个很好的三元循环码族，它包含了一些最著名的线性码，并且与Chen et al.(2023)[2]提出的码相比，该族具有更好的最小距离下界。

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 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-12-01 Epub Date: 2025-07-07 DOI: 10.1016/j.ffa.2025.102691

Kaimin Cheng

Let q be a prime power, and let

F_{q}

denote the finite field with q elements. Consider a positive integer n, and let

R = {R_{i}}_{i = 0}^{n - 1}

be a family of subsets of

F_{q}

. Define

N (R, n)

as the number of monic irreducible polynomials of degree n over

F_{q}

where the coefficient of each non-leading term

T^{i}

lies in

F_{q} ∖ R_{i}

. In this paper, we provide an asymptotic formula for

N (R, n)

, extending a result of Porritt to a more general case.

设q是一个素数幂，设Fq表示有q个元素的有限域。考虑一个正整数n，设R={Ri}i=0n−1是Fq的子集族。定义N（R， N）为N / Fq次的不可约一元多项式的个数，其中每个非前导项Ti的系数在Fq∈Ri中。本文给出了N（R， N）的渐近公式，将Porritt的结果推广到更一般的情况。

引用次数: 0

Partial difference sets from unions of cyclotomic classes 分环类并集的偏差集

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-12-01 Epub Date: 2025-06-02 DOI: 10.1016/j.ffa.2025.102661

Ka Hin Leung , Koji Momihara , Qing Xiang

In their study of two-weight irreducible cyclic codes, Schmidt and White (2002) obtained a necessary and sufficient condition on

(q, N)

under which the multiplicative subgroup of index N of the finite field

F_{q}

forms a regular partial difference set (PDS) in the additive group of

F_{q}

. They also found 11 sporadic examples by a computer search aside from two known infinite families of PDS. In this paper, we study the problem of determining for which

(q, N)

a union of multiple cosets of the multiplicative subgroup of index N of

F_{q}

forms a regular PDS in the additive group of

F_{q}

. Building on the work of Schmidt and White, we find a necessary and sufficient numerical condition on the parameters

(q, N)

for unions of multiple cyclotomic classes to form regular PDS in

(F_{q}, +)

. We then apply the theorem to the situation where unions of a small number of classes are selected in a structured manner. We obtain a new infinite family of regular PDS not belonging to previously known families, and two sporadic examples of regular PDS (one of which is new) with the help of a computer research. We further propose a conjecture analogous to the Schmidt-White conjecture proposed in their 2002 paper.

Schmidt和White（2002）在二权不可约循环码的研究中，得到了有限域Fq的指标N的乘法子群在Fq的可加群中形成正则偏差分集（PDS）的一个充要条件（q，N）。除了两个已知的无限家族外，他们还通过计算机搜索发现了11个零星的例子。本文研究了判定Fq的指标N的乘法子群的多个余集的并集在Fq的加性群中是否形成正则PDS的问题。在Schmidt和White工作的基础上，我们找到了多个环切分类的联合在（Fq,+）中形成正则PDS的参数（q，N）的一个充要条件。然后，我们将该定理应用于以结构化方式选择少量类的联合的情况。在计算机研究的帮助下，我们得到了一个新的不属于已知族的正则PDS无限族，以及两个零星的正则PDS例子（其中一个是新的）。我们进一步提出了一个类似于他们2002年论文中提出的Schmidt-White猜想的猜想。

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引用次数: 0

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

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-12-01 Epub Date: 2025-07-07 DOI: 10.1016/j.ffa.2025.102687

Nuno Arala , Sam Chow

We establish expansion properties for suitably generic polynomials of degree d in

d + 1

variables over finite fields. In particular, we show that if

P \in F_{q} [x_{1}, \dots, x_{d + 1}]

is a polynomial of degree d, whose coefficients avoid the zero locus of some explicit polynomial of degree

O_{d} (1)

, and

X_{1}, \dots, X_{d + 1} \subseteq F_{q}

are suitably large, then

| P (X_{1}, \dots, X_{d + 1}) | = q - O (1)

. Our methods rely on a higher-degree extension of a result of Vinh on point–line incidences over a finite field.

建立了有限域上d+1变量下d次多项式的适当泛型展开性质。特别地，我们证明了如果P∈Fq[x1，…，xd+1]是一个d次多项式，其系数避开某Od(1)次显式多项式的零轨迹，且x1，…，xd+1适宜大，则|P(x1，…，xd+1)|=q−O(1)。我们的方法依赖于Vinh关于有限域上点线关联的结果的更高次推广。

引用次数: 0

Decoding up to Hartmann–Tzeng and Roos bounds for rank codes 解码到Hartmann-Tzeng和Roos界的秩码

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-12-01 Epub Date: 2025-06-11 DOI: 10.1016/j.ffa.2025.102676

José Manuel Muñoz

A class of linear block codes which simultaneously generalizes Gabidulin codes and a class of skew cyclic codes is defined. For these codes, both a Hartmann–Tzeng-like bound and a Roos-like bound, with respect to their rank distance, are described, and corresponding nearest-neighbor decoding algorithms are presented. Additional necessary conditions so that decoding can be done up to the described bounds are studied. Subfield subcodes and interleaved codes from the considered class of codes are also described, since they allow an unbounded length for the codes, providing a decoding algorithm for them; additionally, both approaches are shown to yield equivalent codes with respect to the rank metric.

定义了一类同时推广加比度林码和一类偏循环码的线性分组码。对于这些码，分别描述了关于秩距离的类hartmann - tzeng_bound和类roos_bound，并给出了相应的最近邻解码算法。另外的必要条件，使解码可以完成到所描述的界限进行了研究。还描述了所考虑的代码类中的子域子码和交错码，因为它们允许代码的无界长度，并为它们提供了解码算法；此外，两种方法都显示出相对于秩度量产生等效的代码。

引用次数: 0

Deep holes of twisted Reed-Solomon codes 扭曲的里德-所罗门密码的深洞

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-12-01 Epub Date: 2025-06-16 DOI: 10.1016/j.ffa.2025.102680

Weijun Fang , Jingke Xu , Ruiqi Zhu

The deep holes of a linear code are the vectors that achieve the maximum error distance (covering radius) to the code. Determining the covering radius and deep holes of linear codes is a fundamental problem in coding theory. In this paper, we investigate the problem of deep holes of twisted Reed-Solomon codes. The covering radius and a standard class of deep holes of twisted Reed-Solomon codes

{TRS}_{k} (A, θ)

are obtained for a general evaluation set

A \subseteq F_{q}

. Furthermore, we consider the problem of determining all deep holes of the full-length twisted Reed-Solomon codes

{TRS}_{k} (F_{q}, θ)

. For even q, by utilizing the polynomial method and Gauss sums over finite fields, we prove that the standard deep holes are all the deep holes of

{TRS}_{k} (F_{q}, θ)

with

\frac{3 q - 4}{4} \leq k \leq q - 4

. For odd q, we adopt a different method and employ the results on some equations over finite fields to show that there are also no other deep holes of

{TRS}_{k} (F_{q}, θ)

with

\frac{3 q + 3 \sqrt{q} - 7}{4} \leq k \leq q - 4

. In addition, for the boundary cases of

k = q - 3, q - 2

and

q - 1

, we completely determine their deep holes using results on certain character sums.

线性码的深孔是达到码的最大误差距离（覆盖半径）的向量。确定线性码的覆盖半径和深孔是编码理论中的一个基本问题。本文研究了扭曲Reed-Solomon码的深孔问题。对一般评价集a， Fq，得到了扭曲Reed-Solomon码TRSk（a,θ）的覆盖半径和深孔标准类。进一步研究了长度扭曲Reed-Solomon码TRSk（Fq,θ）的所有深孔的确定问题。对于偶q，利用多项式方法和有限域上的高斯和证明了标准深孔都是3q−44≤k≤q−4的TRSk（Fq,θ）的深孔。对于奇数q，我们采用了不同的方法，并利用有限域上的一些方程的结果证明了TRSk（Fq,θ）也不存在3q+3q−74≤k≤q−4的其他深孔。此外，对于k=q−3、q−2和q−1的边界情况，我们利用某些特征和的结果完全确定了它们的深孔。

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引用次数: 0

Projective symplectic groups of genus one and two 一属和二属的射影辛群

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-12-01 Epub Date: 2025-07-07 DOI: 10.1016/j.ffa.2025.102689

Rezhna M. Hussein, Haval M. Mohammed Salih

In this article, we provide a comprehensive classification of all primitive genus one and genus two systems of the finite group G with

P S p (4, q) \leq G \leq A u t (P S p (4, q))

, where q is a prime power. Also, we use computational tools to show that G possesses no genus g group if

q > 5

where

g = 1

, and 2.

本文给出了有限群G中PSp(4,q)≤G≤Aut（PSp(4,q)）的所有原始一格和二格系统的一个综合分类，其中q是素幂。此外，我们使用计算工具证明，如果G =1和2，则G不具有G属群。

引用次数: 0

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全部 ACS Publications Elsevier ieeexplore Springer The Royal Society of Chemistry Wiley

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Finite Fields and Their Applications

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