Finite Fields and Their Applications最新文献

英文中文

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

Euclidean sets with only one distance modulo a prime ideal 只有一个距离模素理想的欧几里得集合

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-07-03 DOI: 10.1016/j.ffa.2025.102690

Hiroshi Nozaki

<div><div>Let X be a finite set in the Euclidean space <math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math>. If the squared distance between any two distinct points in X is an odd integer, then the cardinality of X is bounded above by <math><mi>d</mi><mo>+</mo><mn>2</mn></math>, as shown by Rosenfeld (1997) or Smith (1995). They proved that there exists a <math><mo>(</mo><mi>d</mi><mo>+</mo><mn>2</mn><mo>)</mo></math>-point set X in <math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math> having only odd integral squared distances if and only if <math><mi>d</mi><mo>+</mo><mn>2</mn></math> is congruent to 0 modulo 4. The distances can be interpreted as an element of the finite field <math><mi>Z</mi><mo>/</mo><mn>2</mn><mi>Z</mi></math>. We generalize this result for a local ring <math><mo>(</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>,</mo><mi>p</mi><msub><mrow><mi>A</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>)</mo></math> as follows. Let K be an algebraic number field that can be embedded into <math><mi>R</mi></math>. Fix an embedding of K into <math><mi>R</mi></math>, and K is interpreted as a subfield of <math><mi>R</mi></math>. Let <math><mi>A</mi><mo>=</mo><msub><mrow><mi>O</mi></mrow><mrow><mi>K</mi></mrow></msub></math> be the ring of integers of K, and <math><mi>p</mi></math> a prime ideal of <math><msub><mrow><mi>O</mi></mrow><mrow><mi>K</mi></mrow></msub></math>. Let <math><mo>(</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>,</mo><mi>p</mi><msub><mrow><mi>A</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>)</mo></math> be the local ring obtained from the localization <math><msup><mrow><mo>(</mo><mi>A</mi><mo>∖</mo><mi>p</mi><mo>)</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mi>A</mi></math>, which is interpreted as a subring of <math><mi>R</mi></math>. If the squared distances of <math><mi>X</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math> are in <math><msub><mrow><mi>A</mi></mrow><mrow><mi>p</mi></mrow></msub></math> and each squared distance is congruent to some constant <math><mi>k</mi><mo>≢</mo><mn>0</mn></math> modulo <math><mi>p</mi><msub><mrow><mi>A</mi></mrow><mrow><mi>p</mi></mrow></msub></math>, then <math><mo>|</mo><mi>X</mi><mo>|</mo><mo>≤</mo><mi>d</mi><mo>+</mo><mn>2</mn></math>, as shown by Nozaki (2023). In this paper, we prove that there exists a set <math><mi>X</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math> attaining the upper bound <math><mo>|</mo><mi>X</mi><mo>|</mo><mo>≤</mo><mi>d</mi><mo>+</mo><mn>2</mn></math>

设X是欧几里德空间Rd中的一个有限集合。如果X中任意两个不同点之间的距离的平方是奇数，则X的基数以d+2为界，如Rosenfeld（1997）或Smith（1995）所示。他们证明了在Rd中存在一个（d+2）点集X只有奇数的平方积分距离当且仅当d+2等于0取4模。距离可以解释为有限场Z/2Z的一个元素。我们将这个结果推广到局部环（Ap,pAp）如下：设K是一个可嵌入R中的代数数域，将K嵌入R，则K被解释为R的一个子域。设a =OK是K的整数环，p是OK的素理想。设（Ap,pAp）是由定位(A∈p)−1A得到的局部环，它被解释为r的子环。如果X∧Rd的平方距离在Ap中，并且每个平方距离都等于某个常数k 0模pAp，则|X|≤d+2，如Nozaki（2023）所示。本文证明了存在一个集合X∧Rd，当且仅当当有限域Ap/pAp为特征2时，d+2与0模4全等，且当Ap/pAp为特征p奇时，d+2与0模p全等，达到上界|X|≤d+2。我们也提供了得到这个上界的例子。

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 : 2025-06-30 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

Extending a result of Carlitz and McConnel to polynomials which are not permutations 将Carlitz和McConnel的结果推广到非置换多项式

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-06-27 DOI: 10.1016/j.ffa.2025.102683

Bence Csajbók

Let D denote the set of directions determined by the graph of a polynomial f of

F_{q} [x]

, where q is a power of the prime p. If D is contained in a multiplicative subgroup M of

F_{q}^{\times}

, then by a result of Carlitz and McConnel it follows that

f (x) = a x^{p^{k}} + b

for some

k \in N

. Of course, if

D \subseteq M

, then

0 \notin D

and hence f is a permutation. If we assume the weaker condition

D \subseteq M \cup {0}

, then f is not necessarily a permutation, but Sziklai conjectured that

f (x) = a x^{p^{k}} + b

follows also in this case. When q is odd, and the index of M is even, then a result of Ball, Blokhuis, Brouwer, Storme and Szőnyi combined with a result of Göloğlu and McGuire proves the conjecture. Assume

\deg f \geq 1

. We prove that if the size of

D^{- 1} D = {d^{- 1} d^{'} : d \in D ∖ {0}, d^{'} \in D}

is less than

q - \deg f + 2

, then f is a permutation of

F_{q}

. We use this result to prove the conjecture of Sziklai.

设D表示由Fq[x]的多项式f的图确定的方向集，其中q是素数p的幂。如果D包含在fqx的乘法子群M中，则根据Carlitz和McConnel的结果，可以得出对于k∈N， f(x)=axpk+b。当然，若D∈M，则0∈D，故f是一个置换。如果我们假设弱条件D≥M∪{0}，则f不一定是一个置换，但Sziklai推测在这种情况下f(x)=axpk+b也成立。当q为奇数，M的指标为偶数时，Ball、Blokhuis、browwer、Storme和Szőnyi的结果结合Göloğlu和McGuire的结果证明了猜想。假设度⁡f≥1。证明了如果D−1D的大小={D−1D ': D∈D∈{0},D '∈D}小于q−deg (f+2)，则f是Fq的一个置换。我们用这个结果证明了Sziklai的猜想。

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引用次数: 0

Some new classes of permutation polynomials and their compositional inverses 几种新的置换多项式及其复合逆

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-06-27 DOI: 10.1016/j.ffa.2025.102685

Sartaj Ul Hasan, Ramandeep Kaur

We focus on permutation polynomials of the form

L (X) + {Tr}_{m}^{3 m} {(X)}^{s}

over

F_{q^{3}}

, where

F_{q}

is the finite field with

q = p^{m}

elements, p is a prime number, m is a positive integer,

{Tr}_{m}^{3 m} (\cdot)

is the relative trace function from

F_{p^{3 m}}

F_{p^{m}}

L (X)

is a linearized polynomial over

F_{q^{3}}

, and

s > 1

is a positive integer. More precisely, we present six new classes of permutation polynomials over

F_{q^{3}}

of the aforementioned form: one class over finite fields of even characteristic, three classes over finite fields of odd characteristic, and the remaining two over finite fields of arbitrary characteristic. Furthermore, we show that these classes of permutation polynomials are inequivalent to the known ones of the same form. We also provide explicit expressions for the compositional inverses of each of these classes of permutation polynomials.

我们重点研究了L(X)+Trm3m(X)s / Fq3的置换多项式，其中Fq是具有q=pm元素的有限域，p是素数，m是正整数，Trm3m（⋅）是Fp3m到Fpm的相对迹函数，L(X)是Fq3上的线性化多项式，s>；1是正整数。更准确地说，我们提出了上述形式的Fq3上六类新的置换多项式：一类是偶特征有限域上的置换多项式，三类是奇特征有限域上的置换多项式，其余两类是任意特征有限域上的置换多项式。进一步，我们证明了这类置换多项式与已知的相同形式的置换多项式是不等价的。我们还提供了这类置换多项式的组合逆的显式表达式。

引用次数: 0

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