Finite Fields and Their Applications最新文献

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Square values of several polynomials over a finite field 有限域上若干多项式的平方值

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-07-11 DOI: 10.1016/j.ffa.2025.102696

Kaloyan Slavov

Let

f_{1}, \dots, f_{m}

be polynomials in n variables with coefficients in a finite field

F_{q}

. We estimate the number of points

\underline{x}

F_{q}^{n}

such that each value

f_{i} (\underline{x})

is a nonzero square in

F_{q}

. The error term is especially small when the

f_{i}

define smooth projective quadrics with nonsingular intersections. We improve the error term in a recent work by Asgarli–Yip on mutual position of smooth quadrics.

设f1，…，fm是有限域Fq中n个变量的多项式。我们估计Fqn中x_的个数，使得每个值fi（x_）是Fq中的非零平方。当定义具有非奇异交点的光滑投影二次曲面时，误差项特别小。在Asgarli-Yip最近关于光滑二次曲面互位的研究中，我们改进了误差项。

引用次数: 0

Bounds on the size of (r,δ)-locally repairable codes for fixed values q and d （r,δ）-定值q和d的局部可修码的大小界限

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-07-11 DOI: 10.1016/j.ffa.2025.102697

Wenqin Zhang , Chen Yuan , Yuan Luo , Nian Li

Erasure codes can strengthen fault-tolerance and reliability in distributed storage systems. One of these, locally repairable codes (LRCs), plays a crucial role. A locally repairable code with locality r (called r-LRC) can recover any coded symbol by accessing at most r other coded symbols. The original definition of LRCs is used to repair a single failed node. To address the practical concern of multiple node failures, the concept of LRCs with locality

(r, δ)

was introduced by Prakash et al. (2012) which can be seen as a generalization of r-LRCs. Since then, the bounds and constructions of

(r, δ)

-LRCs have been extensively studied.

This paper is dedicated to both the bounds and constructions of

(r, δ)

-LRCs with disjoint local repair groups over a finite field

F_{q}

, where the parameters r, δ, the alphabet size q, and the minimum distance d are fixed constants, while the code length n tends to infinity. Inspired by the method of the classical Gilbert-Varshamov (GV) bound, we first derive an asymptotic Gilbert-Varshamov-type bound for

(r, δ)

-LRCs in this regime. We manage to show that this GV-type bound works as a threshold for random linear

(r, δ)

-LRCs with disjoint local repair groups using the first and second moment methods. As a corollary, such a random linear

(r, δ)

-LRC has a high probability of attaining the GV-type bound. As an analogue to the classic GV-type bound, we present two constructions of

(r, δ)

-LRCs that each beats this GV-type bound. One construction is obtained from a straightforward concatenation of an outer BCH code and an inner

[r + δ - 1, r]

MDS code. Another construction is based on the Kronecker product of two matrices. To complement our results, the case that n is large but finite is also considered. In this regime, we provide an explicit upper bound for the binary r-LRCs, which is an improvement over the one in Ma and Ge (2019). Furthermore, this bound is shown to be tight for some specific parameters.

Erasure code可以增强分布式存储系统的容错性和可靠性。其中，局部可修复编码（lrc）起着至关重要的作用。局部性为r的局部可修复码（称为r- lrc）可以通过访问最多r个其他编码符号来恢复任何编码符号。lrc的原始定义用于修复单个故障节点。为了解决多节点故障的实际问题，Prakash等人（2012）引入了局域性（r,δ）的lrc概念，可以看作是r- lrc的推广。从那时起，(r,δ)- lrc的界和结构得到了广泛的研究。本文研究有限域Fq上具有不相交局部修群的(r,δ)- lrc的界和构造，其中参数r，δ，字母大小q和最小距离d是固定常数，而编码长度n趋于无穷。在经典Gilbert-Varshamov （GV）界方法的启发下，我们首先导出了(r,δ)- lrc的渐近Gilbert-Varshamov型界。我们使用第一和第二矩方法成功地证明了这种gv型边界作为具有不接合的局部修复群的随机线性(r,δ)- lrc的阈值。作为一个推论，这样的随机线性(r,δ)-LRC有很高的概率达到gv型界。作为经典gv型界的类似物，我们提出了(r,δ)- lrc的两种结构，它们都优于该gv型界。一种结构是由外部BCH码和内部[r+δ−1，r] MDS码的直接连接得到的。另一种构造是基于两个矩阵的克罗内克积。为了补充我们的结果，我们还考虑了n很大但有限的情况。在这种情况下，我们为二元r- lrc提供了明确的上限，这是对Ma和Ge（2019）中的上限的改进。此外，对于某些特定参数，该界是紧的。

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引用次数: 0

New constructions of permutation polynomials of the form x+γTrqq3(h(x)) over finite fields with even characteristic 偶特征有限域上形式为x+γTrqq3（h(x)）的置换多项式的新构造

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-07-09 DOI: 10.1016/j.ffa.2025.102694

Sha Jiang, Kangquan Li, Longjiang Qu

Permutation polynomials over finite fields have applications in many areas of mathematics and engineering. Particularly, permutation polynomials of the form

x + γ {Tr}_{q}^{q^{n}} (h (x))

have been studied for a long time. In this paper, we further investigate permutation polynomials of the form

x + γ {Tr}_{q}^{q^{3}} (h (x))

over finite fields with even characteristic. For one thing, by choosing functions

h (x)

with a low q-degree, we propose four classes of permutation polynomials of the form

x + γ {Tr}_{q}^{q^{3}} (h (x))

over

F_{q^{3}}

. For the other thing, we give seven classes of permutations of the form

x + {Tr}_{q}^{q^{3}} (h (x))

with binomials

h (x)

over

F_{q^{3}}

. Finally, we also show that the permutation polynomials constructed in this paper are not quasi-multiplicative equivalent to the known permutation polynomials.

有限域上的置换多项式在数学和工程的许多领域都有应用。特别是形式为x+γTrqqn（h(x)）的置换多项式已经被研究了很长时间。本文进一步研究了具有偶特征的有限域上形式为x+γTrqq3（h(x)）的置换多项式。一方面，通过选择低q度的函数h(x)，我们提出了四类形式为x+γTrqq3(h(x)) / Fq3的置换多项式。另一方面，我们给出了7种形式为x+Trqq3（h(x)）的二项式h(x) / Fq3的排列。最后，我们还证明了本文构造的置换多项式与已知的置换多项式不是拟乘法等价的。

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 : 2025-07-07 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

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 : 2025-07-07 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

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 : 2025-07-07 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

A new construction of maximally recoverable codes with hierarchical locality 一种具有分层局部性的最大可恢复码的新构造

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-07-07 DOI: 10.1016/j.ffa.2025.102686

Rajendra Prasad Rajpurohit, Maheshanand Bhaintwal

In this paper, we present a novel construction of maximally recoverable codes with two-level hierarchical locality using a parity-check matrix approach. The construction given in this paper utilizes Gabidulin codes for mid-level heavy parities and linearized Reed-Solomon codes for global heavy parities. When the number of local sets is small, this construction performs better than the previously known constructions as the field size required in our construction is smaller for such cases, making it useful for practical scenarios in distributed data storage systems. We also consider a special case of our construction when the number of global parities is fixed and is equal to 1. In this case, our construction performs better when the number of local sets is small and the number of mid-level parities is even.

本文利用奇偶校验矩阵的方法，提出了一种具有两级分层局部性的最大可恢复码的构造方法。本文给出的构造方法使用Gabidulin码表示中级重偶，线性化Reed-Solomon码表示全局重偶。当局部集的数量很少时，这种构造比以前已知的构造表现得更好，因为在这种情况下，我们的构造所需的字段大小更小，这使得它对分布式数据存储系统中的实际场景很有用。我们还考虑了我们的构造的一个特殊情况，即全局奇偶的数量是固定的并且等于1。在这种情况下，我们的构造在局部集的数量较少且中级奇偶的数量为偶数时表现更好。

引用次数: 0

On the set of stable primes for postcritically infinite maps over number fields 数域上后临界无限映射的稳定素数集

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-07-07 DOI: 10.1016/j.ffa.2025.102693

Joachim König

Many interesting questions in arithmetic dynamics revolve, in one way or another, around the (local and/or global) reducibility behavior of iterates of a polynomial. We show that for very general families of integer polynomials f (and, more generally, rational functions over number fields), the set of stable primes, i.e., primes modulo which all iterates of f are irreducible, is a density zero set. Compared to previous results, our families cover a much wider ground, and in particular apply to 100% of polynomials of any given odd degree, thus adding evidence to the conjecture that polynomials with a “large” set of stable primes are necessarily of a very specific shape, and in particular are necessarily postcritically finite.

算术动力学中许多有趣的问题以某种方式围绕多项式迭代的（局部和/或全局）可约性行为。我们证明了对于非常一般的整数多项式族f（以及更一般的数域上的有理函数），稳定素数的集合，即所有迭代f都不可约的素数模，是一个密度零集。与以前的结果相比，我们的家族涵盖了更广泛的领域，特别是适用于任何给定奇数次的多项式的100%，从而为猜想提供了证据，即具有“大”稳定素数集的多项式必然具有非常特定的形状，特别是必然是后临界有限的。

引用次数: 0

Generalizing blocking semiovals in finite projective planes 有限投影平面上块半椭圆的推广

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-07-07 DOI: 10.1016/j.ffa.2025.102688

Marilena Crupi , Antonino Ficarra

Blocking semiovals and the determination of their (minimum) sizes constitute one of the central research topics in finite projective geometry. In this article we introduce the concept of blocking set with the

r_{\infty}

-property in a finite projective plane

PG (2, q)

, with

r_{\infty}

a line of

PG (2, q)

and q a prime power. This notion greatly generalizes that of blocking semioval. We address the question of determining those integers k for which there exists a blocking set of size k with the

r_{\infty}

-property. To solve this problem, we build new theory which deeply analyzes the interplay between blocking sets in finite projective and affine planes.

块半椭圆及其（最小）尺寸的确定是有限射影几何研究的中心课题之一。本文引入了有限射影平面PG（2,q）上具有r∞-性质的块集的概念，其中r∞是PG（2,q）的一条直线，q是素数幂。这个概念极大地推广了阻塞半进程的概念。我们讨论了确定存在大小为k且具有r∞性质的块集的整数k的问题。为了解决这一问题，我们建立了新的理论，深入分析了有限射影平面和仿射平面上的块集之间的相互作用。

引用次数: 0

Decoding algorithms in group codes 组码译码算法

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-07-07 DOI: 10.1016/j.ffa.2025.102692

C. Martínez, F. Molina, A. Piñera-Nicolás

Group codes are linear codes that can be identified with (two-sided) ideals of a group algebra

K G

. Assuming that

K G

is semisimple, we use its decomposition as the direct sum of two ideals, one of them the group code, to design two decoding algorithms. The first one generalizes Meggitt's algorithm designed for cyclic codes, while the other one is inspired in the decoding algorithm studied in [10] and aims to improve it.

群码是可以用群代数KG的（双面）理想识别的线性码。假设KG是半简单的，我们将其分解为两个理想的直接和，其中一个是群码，来设计两种解码算法。第一个是对循环码设计的Meggitt算法的推广，另一个是受到[10]研究的译码算法的启发，旨在对其进行改进。

引用次数: 0

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