Finite Fields and Their Applications最新文献

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The R-transform as power map and its generalizations to higher degree 作为幂图的 R 变换及其对更高程度的概括

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-11-27 DOI: 10.1016/j.ffa.2024.102546

Alp Bassa , Ricardo Menares

We give iterative constructions for irreducible polynomials over

F_{q}

of degree

n \cdot t^{r}

for all

r \geq 0

, starting from irreducible polynomials of degree n. The iterative constructions correspond modulo fractional linear transformations to compositions with power functions

x^{t}

. The R-transform introduced by Cohen is recovered as a particular case corresponding to

x^{2}

, hence we obtain a generalization of Cohen's R-transform (

t = 2

) to arbitrary degrees

t \geq 2

. Important properties like self-reciprocity and invariance of roots under certain automorphisms are deduced from invariance under multiplication by appropriate roots of unity. Extending to quadratic extensions of

F_{q}

we recover and generalize a recursive construction of Panario, Reis and Wang.

我们从 n 度的不可还原多项式出发，给出了所有 r≥0 的 n⋅tr Fq 上不可还原多项式的迭代构造。科恩引入的 R 变换作为与 x2 相对应的特殊情况被复原，因此我们得到了科恩 R 变换 (t=2) 对任意度 t≥2 的推广。从与适当的合一根相乘的不变性推导出了自还原性和根在某些自动形态下的不变性等重要性质。扩展到 Fq 的二次扩展，我们恢复并推广了帕纳里奥、雷斯和王的递归构造。

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引用次数: 0

Webs and squabs of conics over finite fields 有限域上圆锥的网和方锥

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-11-26 DOI: 10.1016/j.ffa.2024.102544

Nour Alnajjarine , Michel Lavrauw

This paper is a contribution towards a solution for the longstanding open problem of classifying linear systems of conics over finite fields initiated by L. E. Dickson in 1908, through his study of the projective equivalence classes of pencils of conics in

PG (2, q)

, for q odd. In this paper a set of complete invariants is determined for the projective equivalence classes of webs and of squabs of conics in

PG (2, q)

, both for q odd and even. Our approach is mainly geometric, and involves a comprehensive study of the geometric and combinatorial properties of the Veronese surface in

PG (5, q)

. The main contribution is the determination of the distribution of the different types of hyperplanes incident with the K-orbit representatives of points and lines of

PG (5, q)

, where

K ≅ PGL (3, q)

, is the subgroup of

PGL (6, q)

stabilizing the Veronese surface.

本文是对 L. E. Dickson 于 1908 年通过研究 PG(2,q) 中圆锥的铅笔的投影等价类（q 为奇数）而提出的有限域上圆锥的线性系统分类这一长期未决问题的一个解决方案的贡献。本文为 PG(2,q) 中圆锥的网状和方形的投影等价类确定了一组完整的不变式，无论是 q 为奇数还是偶数。我们的方法主要是几何方法，涉及对 PG(5,q) 中维罗尼斯曲面的几何和组合性质的全面研究。我们的主要贡献是确定了与 PG(5,q) 的 K 轨道代表点和线相关的不同类型超平面的分布，其中 K≅PGL(3,q) 是 PGL(6,q) 的子群，稳定了 Veronese 曲面。

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引用次数: 0

Some q-ary constacyclic BCH codes with length qm+12 长度为 qm+12 的一些 qary 常环 BCH 码

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-11-21 DOI: 10.1016/j.ffa.2024.102545

Jin Li , Huilian Zhu , Shixin Zhu

In this paper, we study some q-ary λ-constacyclic BCH codes of length

\frac{q^{m} + 1}{2}

with some large designed distances for

ord (λ) = 2

and

2 + ⌈ \frac{4}{q - 3} ⌉ \leq ord (λ) \leq q - 1

respectively, where q is an odd prime power and

ord (λ) | (q - 1)

. The dimensions and the lower bounds on the minimum distances of these codes are given by using recurrence relations and the introduced definitions of sequences. The code examples presented in this paper indicate that these codes have good parameters in general.

本文研究了一些长度为 qm+12 的 qary λ-constacyclic BCH 码，它们在 ord(λ)=2 和 2+⌈4q-3⌉≤ord(λ)≤q-1 时分别具有一些较大的设计距离，其中 q 是奇素数幂且 ord(λ)|(q-1)。利用递推关系和引入的序列定义，给出了这些编码的维数和最小距离下限。本文介绍的编码示例表明，这些编码一般具有良好的参数。

引用次数: 0

Repeated-root constacyclic codes of length kslmpn over finite fields 有限域上长度为 kslmpn 的重复根常环码

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-11-14 DOI: 10.1016/j.ffa.2024.102542

Qi Zhang , Weiqiong Wang , Shuyu Luo , Yue Li

For different odd primes k, l, p, and positive integers s, m, n, the polynomial

x^{k^{s} l^{m} p^{n}} - λ

F_{q} [x]

is explicitly factorized, where p is the characteristic of

F_{q}

λ \in F_{q}^{⁎}

. All repeated-root constacyclic codes and their dual codes of length

k^{s} l^{m} p^{n}

over

F_{q}

are characterized. In addition, the characterization and enumeration of all linear complementary dual (LCD) cyclic and negacyclic codes of length

k^{s} l^{m} p^{n}

over

F_{q}

are obtained.

对于不同的奇数素数 k、l、p 和正整数 s、m、n，Fq[x] 中的多项式 xkslmpn-λ 被显式因式分解，其中 p 是 Fq 的特征，λ∈Fq⁎。表征了 Fq 上长度为 kslmpn 的所有重复根常环码及其对偶码。此外，还得到了 Fq 上所有长度为 kslmpn 的线性互补对偶（LCD）循环码和负循环码的特征和枚举。

引用次数: 0

Complete description of measures corresponding to Abelian varieties over finite fields 有限域上阿贝尔变种对应度量的完整描述

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-11-14 DOI: 10.1016/j.ffa.2024.102543

Nikolai S. Nadirashvili , Michael A. Tsfasman

We study probability measures corresponding to families of abelian varieties over a finite field. These measures play an important role in the Tsfasman–Vlăduţ theory of asymptotic zeta-functions defining completely the limit zeta-function of the family. J.-P. Serre, using results of R.M. Robinson on conjugate algebraic integers, described the possible set of measures than can correspond to families of abelian varieties over a finite field. The problem whether all such measures actually occur was left open. Moreover, Serre supposed that not all such measures correspond to abelian varieties (for example, the Lebesgue measure on a segment). Here we settle Serre's problem proving that Serre conditions are sufficient, and thus describe completely the set of measures corresponding to abelian varieties.

我们研究与有限域上的无性变体族相对应的概率度量。这些度量在完全定义族的极限zeta函数的渐近zeta函数的Tsfasman-Vlăduţ理论中起着重要作用。J.-P.塞雷（J.-P. Serre）利用罗宾逊（R.M. Robinson）关于共轭代数整数的结果，描述了与有限域上无性方程族相对应的可能度量集合。至于是否所有这些度量都会出现，这个问题还没有解决。此外，塞尔认为并非所有这些度量都对应于无比方体（例如，线段上的勒贝格度量）。在这里，我们解决了塞雷的问题，证明塞雷条件是充分的，从而完整地描述了对应于无比方体的度量集合。

引用次数: 0

Self-dual 2-quasi negacyclic codes over finite fields 有限域上的自偶 2-quasi 负环码

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-11-07 DOI: 10.1016/j.ffa.2024.102541

Yun Fan, Yue Leng

In this paper, we investigate the existence and asymptotic properties of self-dual 2-quasi negacyclic codes of length 2n over a finite field of cardinality q. When n is odd, we show that the q-ary self-dual 2-quasi negacyclic codes exist if and only if

q ≢ - 1 (\mod 4)

. When n is even, we prove that the q-ary self-dual 2-quasi negacyclic codes always exist. By using the technique introduced in this paper, we prove that q-ary self-dual 2-quasi negacyclic codes are asymptotically good.

当 n 为奇数时，我们证明了当且仅当 q≢-1(mod4)时 q-ary 自双 2-quasi 负环码存在。当 n 为偶数时，我们证明 q-ary 自双 2-quasi 负环码总是存在的。利用本文介绍的技术，我们证明了 qary 自双 2-quasi 负环码是渐近良好的。

引用次数: 0

Intersecting families of polynomials over finite fields 有限域上多项式的相交族

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-11-07 DOI: 10.1016/j.ffa.2024.102540

Nika Salia , Dávid Tóth

This paper demonstrates an analog of the Erdős–Ko–Rado theorem to polynomial rings over finite fields, affirmatively answering a conjecture of C. Tompkins.

A k-uniform family of subsets of a set of size n is ℓ-intersecting if any two subsets in the family intersect in at least ℓ elements. The study of such intersecting families is a core subject of extremal set theory, tracing its roots to the seminal 1961 Erdős–Ko–Rado theorem, which establishes a sharp upper bound on the size of these families. Here, we extend the Erdős–Ko–Rado theorem to polynomial rings over finite fields.

Specifically, we determine the largest possible size of a family of monic polynomials, each of degree n, over a finite field

F_{q}

, where every pair of polynomials in the family shares a common factor of degree at least ℓ. We prove that the upper bound for this size is

q^{n - ℓ}

and characterize all extremal families that achieve this maximum size.

Extending to triple-intersecting families, where every triplet of polynomials shares a common factor of degree at least ℓ, we prove that only trivial families achieve the corresponding upper bound. Moreover, by relaxing the conditions to include polynomials of degree at most n, we affirm that only trivial families achieve the corresponding upper bound.

本文证明了厄尔多-柯-拉多定理在有限域上多项式环中的类比，肯定地回答了汤普金斯（C. Tompkins）的猜想。如果一个大小为 n 的集合的 k 个均匀子集族中的任意两个子集至少有 ℓ 个元素相交，那么这个子集族就是 ℓ 相交族。对这种相交族的研究是极值集合理论的核心课题，其根源可追溯到 1961 年的开创性厄多-柯-拉多定理，该定理为这些族的大小确定了一个尖锐的上限。在这里，我们将厄尔多斯-柯-拉多定理扩展到有限域上的多项式环。具体地说，我们确定了有限域 Fq 上每个度数为 n 的单项式族的最大可能规模，其中族中的每对多项式都有一个度数至少为 ℓ 的公因子。我们证明了这一大小的上界是 qn-ℓ，并描述了达到这一最大大小的所有极值族的特征。扩展到三重相交族，其中每个三重多项式共享至少 ℓ 阶的公因子，我们证明只有三重族达到了相应的上界。此外，通过将条件放宽到包括阶数至多为 n 的多项式，我们肯定只有三交系达到了相应的上限。

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引用次数: 0

Partial difference sets with Denniston parameters in elementary abelian p-groups 初等无性 p 群中具有丹尼斯顿参数的部分差集

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-11-07 DOI: 10.1016/j.ffa.2024.102539

Jingjun Bao , Qing Xiang , Meng Zhao

<div><div>Denniston <span><span>[12]</span></span> constructed partial difference sets (PDS) with parameters <span><math><mo>(</mo><msup><mrow><mn>2</mn></mrow><mrow><mn>3</mn><mi>m</mi></mrow></msup><mo>,</mo><mo>(</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi><mo>+</mo><mi>r</mi></mrow></msup><mo>−</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup><mo>+</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>r</mi></mrow></msup><mo>)</mo><mo>(</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup><mo>−</mo><mn>1</mn><mo>)</mo><mo>,</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup><mo>−</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>r</mi></mrow></msup><mo>+</mo><mo>(</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi><mo>+</mo><mi>r</mi></mrow></msup><mo>−</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup><mo>+</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>r</mi></mrow></msup><mo>)</mo><mo>(</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>r</mi></mrow></msup><mo>−</mo><mn>2</mn><mo>)</mo><mo>,</mo><mo>(</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi><mo>+</mo><mi>r</mi></mrow></msup><mo>−</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup><mo>+</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>r</mi></mrow></msup><mo>)</mo><mo>(</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>r</mi></mrow></msup><mo>−</mo><mn>1</mn><mo>)</mo><mo>)</mo></math></span> in elementary abelian groups of order <span><math><msup><mrow><mn>2</mn></mrow><mrow><mn>3</mn><mi>m</mi></mrow></msup></math></span> for all <span><math><mi>m</mi><mo>≥</mo><mn>2</mn></math></span> and <span><math><mn>1</mn><mo>≤</mo><mi>r</mi><mo><</mo><mi>m</mi></math></span>. These PDS arise from maximal arcs in the Desarguesian projective planes PG<span><math><mo>(</mo><mn>2</mn><mo>,</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup><mo>)</mo></math></span>. Davis et al. <span><span>[10]</span></span> and also De Winter <span><span>[13]</span></span> presented constructions of PDS with Denniston parameters <span><math><mo>(</mo><msup><mrow><mi>p</mi></mrow><mrow><mn>3</mn><mi>m</mi></mrow></msup><mo>,</mo><mo>(</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>m</mi><mo>+</mo><mi>r</mi></mrow></msup><mo>−</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>+</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msup><mo>)</mo><mo>(</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>−</mo><mn>1</mn><mo>)</mo><mo>,</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>−</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msup><mo>+</mo><mo>(</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>m</mi><mo>+</mo><mi>r</mi></mrow></msup><mo>−</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>+</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msup><mo>)</mo><mo>(</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msup><mo>−</mo><mn>2</mn><mo>)</mo><mo>,</mo><mo>(</mo><msup><mrow><mi>p</mi>

Denniston [12] 构建了参数为 (23m,(2m+r-2m+2r)(2m-1),2m-2r+(2m+r-2m+2r)(2r-2),(2m+r-2m+2r)(2r-1)) 的基本无边际群中阶数为 23m 的局部差集（PDS），对于所有 m≥2 且 1≤r<m 均适用。这些 PDS 源自 Desarguesian 投影平面 PG(2,2m) 中的最大弧。戴维斯等人 [10] 和德温特 [13] 提出了在所有 m≥2 和 r∈{1,m-1} 条件下，阶数为 p3m 的初等无邻群中具有丹尼斯顿参数 (p3m,(pm+r-pm+pr)(pm-1),pm-pr+(pm+r-pm+pr)(pr-2),(pm+r-pm+pr)(pr-1) 的 PDS 的构造，其中 p 是奇素数。Ball, Blokhuis 和 Mazzocca [1] 证明，对于任何奇素数幂 q，PG(2,qm) 中都不存在非奇数最大弧，因此 [10], [13] 中的构造尤其引人入胜。在本文中，我们证明了在所有 m≥2 和 1≤r<m 条件下，阶数为 q3m 的初等无邻群中存在具有丹尼斯顿参数 (q3m,(qm+r-qm+qr)(qm-1),qm-qr+(qm+r-qm+qr)(qr-2),(qm+r-qm+qr)(qr-1)) 的 PDS，其中 q 是任意素幂。

引用次数: 0

Asymptotic distributions of the number of zeros of random polynomials in Hayes equivalence class over a finite field 有限域上 Hayes 等价类中随机多项式零点数的渐近分布

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-11-06 DOI: 10.1016/j.ffa.2024.102524

Zhicheng Gao

Hayes equivalence is defined on monic polynomials over a finite field

F_{q}

in terms of the prescribed leading coefficients and the residue classes modulo a given monic polynomial Q. We study the distribution of the number of zeros in a random polynomial over finite fields in a given Hayes equivalence class. It is well known that the number of distinct zeros of a random polynomial over

F_{q}

is asymptotically Poisson with mean 1. We show that this is also true for random polynomials in any given Hayes equivalence class. Asymptotic formulas are also given for the number of such polynomials when the degree of such polynomials is proportional to q and the degree of Q and the number of prescribed leading coefficients are bounded by

\sqrt{q}

. When

Q = 1

, the problem is equivalent to the study of the distance distribution in Reed-Solomon codes. Our asymptotic formulas extend some earlier results and imply that all words for a large family of Reed-Solomon codes are ordinary, which further supports the well-known Deep-Hole Conjecture.

在有限域 Fq 上的单项式多项式上，海斯等价性是根据规定的前导系数和残差类 modulo 给定的单项式多项式 Q 定义的。众所周知，Fq 上随机多项式的独特零点数是渐近泊松分布，均值为 1。当此类多项式的度数与 q 成正比，且 Q 的度数和规定的前导系数数以 q 为界时，我们还给出了此类多项式数量的渐近公式。当 Q=1 时，问题等同于研究里德-所罗门码中的距离分布。我们的渐近公式扩展了之前的一些结果，并暗示一大系列里德-所罗门码的所有字都是普通的，这进一步支持了著名的深洞猜想。

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引用次数: 0

Quasi-polycyclic and skew quasi-polycyclic codes over Fq Fq 上的准多环码和偏斜准多环码

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-10-31 DOI: 10.1016/j.ffa.2024.102536

Tushar Bag, Daniel Panario

In this research, our focus is on investigating 1-generator right quasi-polycyclic (QPC) codes over fields. We provide a detailed description of how linear codes with substantial minimum distances can be constructed from QPC codes. We analyze dual QPC codes under various inner products and use them to construct quantum error-correcting codes. Furthermore, our research includes a dedicated section that delves into the area of skew quasi-polycyclic (SQPC) codes, investigating their properties and the role of generators in their construction. This section expands our study to encompass the intriguing area of SQPC codes, offering insights into the non-commutative version of QPC codes, their characteristics and generator structures. Our work deals with the structural properties of QPC, skew polycyclic and SQPC codes, shedding light on their potential for enhancing the field of coding theory.

在这项研究中，我们的重点是研究域上的单生成器右准多环（QPC）码。我们详细描述了如何从 QPC 码中构造出具有可观最小距离的线性码。我们分析了各种内积下的对偶 QPC 码，并用它们来构造量子纠错码。此外，我们的研究还包括一个专门章节，深入探讨偏斜准多环（SQPC）码领域，研究它们的特性以及生成器在其构造中的作用。本节将我们的研究扩展到 SQPC 码这一引人入胜的领域，深入探讨 QPC 码的非交换版本、它们的特性和生成器结构。我们的研究涉及 QPC 码、斜多环码和 SQPC 码的结构特性，揭示了它们在增强编码理论领域的潜力。

引用次数: 0

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