Finite Fields and Their Applications最新文献

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Self-orthogonal cyclic codes with good parameters 具有良好参数的自正交循环码

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-10-31 DOI: 10.1016/j.ffa.2024.102534

Jiayuan Zhang, Xiaoshan Kai, Ping Li

The construction of self-orthogonal codes is an interesting topic due to their wide applications in communication and cryptography. In this paper, we construct several families of self-orthogonal cyclic codes with length

n = \frac{q^{m} - 1}{λ}

, where

λ | q - 1

and

m \geq 3

is odd. It is proved that there exist q-ary self-orthogonal cyclic codes with parameters

[n, \frac{n - 1}{2}, \geq d]

for even prime power q, and

[n, \frac{n}{2} - 1, \geq d]

[n, \frac{n - 1}{2}, \geq d]

for odd prime power q, where d is significantly better than the square-root bound. These several families of self-orthogonal cyclic codes contain some optimal linear codes.

由于自正交码在通信和密码学中的广泛应用，构建自正交码是一个有趣的课题。本文构建了多个长度为 n=qm-1λ（其中 λ|q-1 且 m≥3 为奇数）的自正交循环码族。研究证明，对于偶素数 q，存在参数为 [n,n-12,≥d] 的 qary 自正交循环码；对于奇素数 q，存在参数为 [n,n2-1,≥d] 或 [n,n-12,≥d] 的 qary 自正交循环码，其中 d 明显优于平方根约束。这几个自正交循环码族包含一些最优线性码。

引用次数: 0

Linear codes from planar functions and related covering codes 来自平面函数的线性编码及相关覆盖编码

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-10-31 DOI: 10.1016/j.ffa.2024.102535

Yanan Wu, Yanbin Pan

Linear codes with few weights have wide applications in consumer electronics, data storage system and secret sharing. In this paper, by virtue of planar functions, several infinite families of l-weight linear codes over

F_{p}

are constructed, where l can be any positive integer and p is a prime number. The weight distributions of these codes are determined completely by utilizing certain approach on exponential sums. Experiments show that some (almost) optimal codes in small dimensions can be produced from our results. Moreover, the related covering codes are also investigated.

权重较小的线性编码在消费类电子产品、数据存储系统和秘密共享中有着广泛的应用。本文利用平面函数，构建了多个 Fp 上 l 权重线性编码的无穷族，其中 l 可以是任意正整数，p 是素数。这些编码的权重分布完全是通过利用指数和的某些方法确定的。实验表明，根据我们的结果可以生成一些（几乎）小维度的最优编码。此外，我们还研究了相关的覆盖码。

引用次数: 0

Improvements of the Hasse-Weil-Serre bound over global function fields 全局函数域上哈塞-韦尔-塞雷约束的改进

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-10-31 DOI: 10.1016/j.ffa.2024.102538

Jinjoo Yoo , Yoonjin Lee

We improve the Hasse-Weil-Serre bound over a global function field K with relatively large genus in terms of the ramification behavior of the finite places and the infinite places for

K / k

, where k is the rational function field

F_{q} (T)

. Furthermore, we improve the Hasse-Weil-Serre bound over a global function field K in terms of the defining equation of K. As an application of our main result, we apply our bound to some well-known extensions: Kummer extensions and elementary abelian p-extensions, where p is the characteristic of k. In fact, elementary abelian p-extensions include Artin-Schreier type extensions, Artin-Schreier extensions, and Suzuki function fields. Moreover, we present infinite families of global function fields for Kummer extensions, Artin-Schreier type extensions, and elementary abelian p-extensions but not Artin-Schreier type extensions, which meet our improved bound: our bound is a sharp bound in these families. We also compare our new bound with some known data given in manypoints.org, which is the database on the rational points of algebraic curves. This comparison shows a meaningful improvement of our results on the bound of the number of the rational places of K.

我们从 K/k 的有限位置和无限位置（k 为有理函数域 Fq(T)）的柱化行为出发，改进了具有相对大属的全局函数域 K 上的 Hasse-Weil-Serre 定界。此外，我们还根据 K 的定义方程改进了全局函数域 K 的哈塞-韦尔-塞雷约束：库默扩展和初等无边 p 扩展，其中 p 是 k 的特征。事实上，初等无边 p 扩展包括阿尔丁-施莱尔类型扩展、阿尔丁-施莱尔扩展和铃木函数域。此外，我们还提出了库默扩展、阿廷-施莱尔型扩展和初等常方差 p 扩展的全局函数场无穷族，但不包括阿廷-施莱尔型扩展，它们都符合我们的改进约束：在这些族中，我们的约束是一个尖锐的约束。我们还将我们的新约束与 manypoints.org 中给出的一些已知数据进行了比较，后者是关于代数曲线有理点的数据库。比较结果表明，我们对 K 的有理点数的界值进行了有意义的改进。

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引用次数: 0

On the cyclotomic field Q(e2πi/p) and Zhi-Wei Sun's conjecture on det Mp 关于回旋场 Q(e2πi/p) 和孙志伟关于 det Mp 的猜想

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-10-31 DOI: 10.1016/j.ffa.2024.102533

Li-Yuan Wang , Hai-Liang Wu

In 2019, Zhi-Wei Sun posed an interesting conjecture on certain determinants with Legendre symbol entries. In this paper, by using the arithmetic properties of p-th cyclotomic field and the finite field

F_{p}

, we confirm this conjecture.

2019 年，孙志伟提出了一个关于某些具有 Legendre 符号项的行列式的有趣猜想。本文利用 p-th 回旋域和有限域 Fp 的算术性质，证实了这一猜想。

引用次数: 0

Optimal quinary cyclic codes with three zeros 有三个零的最优二进制循环码

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-10-30 DOI: 10.1016/j.ffa.2024.102537

Jinmei Fan , Xiangyong Zeng

Optimal cyclic codes have received a lot of attention and much progress has been made. However, little is known about optimal quinary cyclic codes. In this paper, by analyzing irreducible factors of certain polynomials over finite fields and utilizing multivariate method, three classes of optimal quinary cyclic codes with parameters

[5^{m} - 1, 5^{m} - 2 m - 2, 4]

and three zeros are presented.

最优循环码受到了广泛关注，并取得了很大进展。然而，人们对最优二元循环码知之甚少。本文通过分析有限域上某些多项式的不可还原因子，并利用多元方法，提出了参数为 [5m-1,5m-2m-2,4] 且有三个零的三类最优二元循环码。

引用次数: 0

On certain maximal curves related to Chebyshev polynomials 论与切比雪夫多项式有关的某些最大曲线

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-10-23 DOI: 10.1016/j.ffa.2024.102521

Guilherme Dias , Saeed Tafazolian , Jaap Top

This paper studies curves defined using Chebyshev polynomials

φ_{d} (x)

over finite fields. Given the hyperelliptic curve

C

corresponding to the equation

v^{2} = φ_{d} (u)

, the prime powers

q \equiv 3 \mod 4

are determined such that

φ_{d} (x)

is separable and

C

is maximal over

F_{q^{2}}

. This extends a result from [30] that treats the special cases

2 | d

as well as d a prime number. In particular a proof of [30, Conjecture 1.7] is presented. Moreover, we give a complete description of the pairs

(d, q)

such that the projective closure of the plane curve defined by

v^{d} = φ_{d} (u)

is smooth and maximal over

F_{q^{2}}

A number of analogous maximality results are discussed.

本文研究在有限域上用切比雪夫多项式 φd(x) 定义的曲线。给定与方程 v2=φd(u) 相对应的超椭圆曲线 C，确定质幂 q≡3mod4 使得 φd(x) 是可分的，且 C 在 Fq2 上是最大的。这扩展了 [30] 中的一个结果，它处理了 2|d 以及 d 是素数的特殊情况。我们特别提出了 [30, 猜想 1.7] 的证明。此外，我们还给出了一对 (d,q) 的完整描述，即 vd=φd(u) 所定义的平面曲线的投影闭包是光滑的，并且是 Fq2 上最大的。

引用次数: 0

New constructions of permutation polynomials of the form x+γTrqq2(h(x)) over finite fields with even characteristic 偶特征有限域上 x+γTrqq2(h(x)) 形式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 : 2024-10-23 DOI: 10.1016/j.ffa.2024.102522

Sha Jiang, Mu Yuan, Kangquan Li, Longjiang Qu

<div><div>Permutation polynomials over finite fields are widely used in cryptography, coding theory, and combinatorial design. Particularly, permutation polynomials of the form <math><mi>x</mi><mo>+</mo><mi>γ</mi><msubsup><mrow><mi>Tr</mi></mrow><mrow><mi>q</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msubsup><mo>(</mo><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo></math> have been studied by many researchers and applied to lift minimal blocking sets. In this paper, we further investigate permutation polynomials of the form <math><mi>x</mi><mo>+</mo><mi>γ</mi><msubsup><mrow><mi>Tr</mi></mrow><mrow><mi>q</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msubsup><mo>(</mo><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo></math> over finite fields with even characteristic. On the one hand, guided by the idea of choosing functions h with a low q-degree, we completely determine the sufficient and necessary conditions of γ for six classes of polynomials of the form <math><mi>x</mi><mo>+</mo><mi>γ</mi><msubsup><mrow><mi>Tr</mi></mrow><mrow><mi>q</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msubsup><mo>(</mo><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo></math> with <math><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>1</mn></mrow></msub><mi>x</mi><mo>+</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>2</mn></mrow></msub><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>3</mn></mrow></msub><msup><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>+</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>4</mn></mrow></msub><msup><mrow><mi>x</mi></mrow><mrow><mi>q</mi><mo>+</mo><mn>2</mn></mrow></msup></math> and <math><msub><mrow><mi>c</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub></math> (<math><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mn>4</mn></math>) to be permutations. These results determine the sizes of directions of these six functions, which is generally difficult. On the other hand, we slightly generalize the above idea and construct other six classes of permutation polynomials of the form <math><mi>x</mi><mo>+</mo><mi>γ</mi><msubsup><mrow><mi>Tr</mi></mrow><mrow><mi>q</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msubsup><mo>(</mo><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo></math> with <math><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>1</mn></mrow></msub><mi>x</mi><mo>+</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>2</mn></mrow></msub><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>3</mn><

有限域上的置换多项式被广泛应用于密码学、编码理论和组合设计中。特别是形式为 x+γTrqqn(h(x)) 的置换多项式已被许多研究人员研究并应用于提升最小阻塞集。本文将进一步研究偶特征有限域上的 x+γTrqq2(h(x))形式的置换多项式。一方面，在选择低 q 阶函数 h 的思想指导下，我们完全确定了六类形式为 x+γTrqq2(h(x))的多项式的 γ 的充分条件和必要条件，其中 h(x)=c1x+c2x2+c3x3+c4xq+2 且 ci∈F2 (i=1,...,4) 为置换。这些结果确定了这六个函数的方向大小，这通常是很困难的。另一方面，我们将上述想法稍作推广，构造了其他六类形式为 x+γTrqq2(h(x)) 的置换多项式，其中 h(x)=c1x+c2x2+c3x3+c4xq+2+x2q-1 和 ci∈F2 (i=1,...,4) 。我们相信，利用这一思想可以得到更多关于 x+γTrqq2(h(x)) 形式的置换多项式的结果。

引用次数: 0

An approach to normal polynomials through symmetrization and symmetric reduction 通过对称化和对称还原实现正多项式的方法

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-10-23 DOI: 10.1016/j.ffa.2024.102525

Darien Connolly , Calvin George , Xiang-dong Hou , Adam Madro , Vincenzo Pallozzi Lavorante

An irreducible polynomial

f \in F_{q} [X]

of degree n is normal over

F_{q}

if and only if its roots

r, r^{q}, \dots, r^{q^{n - 1}}

satisfy the condition

Δ_{n} (r, r^{q}, \dots, r^{q^{n - 1}}) \neq 0

, where

Δ_{n} (X_{0}, \dots, X_{n - 1})

is the

n \times n

circulant determinant. By finding a suitable symmetrization of

Δ_{n}

(A multiple of

Δ_{n}

which is symmetric in

X_{0}, \dots, X_{n - 1}

), we obtain a condition on the coefficients of f that is sufficient for f to be normal. This approach works well for

n \leq 5

but encounters computational difficulties when

n \geq 6

. In the present paper, we consider irreducible polynomials of the form

f = X^{n} + X^{n - 1} + a \in F_{q} [X]

. For

n = 6

and 7, by an indirect method, we are able to find simple conditions on a that are sufficient for f to be normal. In a more general context, we also explore the normal polynomials of a finite Galois extension through the irreducible characters of the Galois group.

当且仅当一个阶数为 n 的不可减多项式 f∈Fq[X] 的根 r,rq,...,rqn-1满足条件 Δn(r,rq,...,rqn-1)≠0，其中 Δn(X0,...,Xn-1)是 n×n 循环行列式时，这个 f∈Fq[X] 在 Fq 上是正常的。通过找到 Δn 的合适对称性（在 X0,...,Xn-1 中对称的 Δn 的倍数），我们就能得到 f 的系数条件，该条件足以保证 f 是正态的。这种方法在 n≤5 时效果很好，但在 n≥6 时遇到了计算上的困难。在本文中，我们考虑 f=Xn+Xn-1+a∈Fq[X] 形式的不可约多项式。对于 n=6 和 7，通过间接方法，我们能够找到关于 a 的简单条件，这些条件足以使 f 成为正多边形。在更一般的情况下，我们还通过伽罗瓦群的不可还原字符来探索有限伽罗瓦扩展的正多项式。

引用次数: 0

The compositional inverses of three classes of permutation polynomials over finite fields 有限域上三类置换多项式的合成逆

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-10-21 DOI: 10.1016/j.ffa.2024.102523

Danyao Wu , Pingzhi Yuan , Huanhuan Guan , Juan Li

R. Gupta, P. Gahlyan and R.K. Sharma presented three classes of permutation trinomials over

F_{q^{3}}

in Finite Fields and Their Applications. In this paper, we employ the local method to prove that those polynomials are indeed permutation polynomials and provide their compositional inverses.

R.Gupta、P. Gahlyan 和 R.K. Sharma 在《有限域及其应用》中提出了三类 Fq3 上的置换三项式。在本文中，我们采用局部方法证明了这些多项式确实是置换多项式，并提供了它们的组成倒数。

引用次数: 0

Stable binomials over finite fields 有限域上的稳定二项式

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-10-18 DOI: 10.1016/j.ffa.2024.102520

Arthur Fernandes , Daniel Panario , Lucas Reis

In this paper, we study stable binomials over finite fields, i.e., irreducible binomials

x^{t} - b \in F_{q} [x]

such that all their iterates are also irreducible over

F_{q}

. We obtain a simple criterion on the stability of binomials based on the forward orbit of 0 under the map

z \mapsto z^{t} - b

. In particular, our criterion extends the one obtained by Jones and Boston (2011) for the quadratic case. As applications of our main result, we obtain an explicit 1-parameter family of stable quartics over prime fields

F_{p}

with

p \equiv 5 (\mod 24)

and also develop an algorithm to test the stability of binomials over finite fields. Finally, building upon a work of Ostafe and Shparlinski (2010), we employ character sums to bound the complexity of such algorithm.

本文研究有限域上的稳定二项式，即不可约二项式 xt-b∈Fq[x]，使得它们的所有迭代也都是 Fq 上的不可约二项式。我们根据 0 在 z↦zt-b 映射下的前向轨道，得到了一个关于二项式稳定性的简单判据。特别是，我们的判据扩展了 Jones 和 Boston（2011）在二次情况下得到的判据。作为我们主要结果的应用，我们得到了质域 Fp 上 p≡5(mod24)的稳定四元数的一个明确的 1 参数族，还开发了一种算法来检验有限域上二项式的稳定性。最后，在 Ostafe 和 Shparlinski（2010 年）工作的基础上，我们利用特征和来约束这种算法的复杂性。

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引用次数: 0

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Finite Fields and Their Applications

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