Finite Fields and Their Applications最新文献

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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

Cyclic locally recoverable LCD codes with the help of cyclotomic polynomials 借助循环多项式的循环局部可恢复液晶编码

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-10-15 DOI: 10.1016/j.ffa.2024.102519

Anuj Kumar Bhagat, Ritumoni Sarma

This article explores two families of cyclic codes over

F_{q}

of length n denoted by

C_{n}

and

C_{n, 1}

, which are generated by the n-th cyclotomic polynomial

Q_{n} (x)

and the polynomial

Q_{n} (x) Q_{1} (x)

, respectively. We find formulae for the distance of

C_{n}

and

C_{n, 1}

for each

n > 1

and conjecture formulae for the distance of their (Euclidean) duals. We prove the conjecture when n is a product of at most two distinct prime powers. Moreover, we show that all these codes are LCD codes, and several subfamilies are both r-optimal and d-optimal locally recoverable codes.

本文探讨了长度为 n 的 Fq 上的两个循环码族，分别用 Cn 和 Cn,1 表示，它们分别由 n 次循环多项式 Qn(x) 和多项式 Qn(x)Q1(x) 生成。我们找到了每个 n>1 的 Cn 和 Cn,1 的距离公式，并猜想了它们（欧几里得）对偶的距离公式。当 n 是最多两个不同质幂的乘积时，我们证明了猜想。此外，我们还证明了所有这些编码都是 LCD 编码，而且有几个子系列既是 r-最优编码，也是 d-最优局部可恢复编码。

引用次数: 0

Some new constructions of optimal and almost optimal locally repairable codes 最优和近似最优局部可修复代码的一些新构造

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-10-15 DOI: 10.1016/j.ffa.2024.102518

Varsha Chauhan, Anuradha Sharma

<div><div>Additive codes over finite fields are natural extensions of linear codes and are useful in constructing quantum error-correcting codes. In this paper, we first study the locality properties of additive MDS codes over finite fields whose dual codes are also MDS. We further provide a method to construct optimal and almost optimal LRCs with new parameters belonging to the family of additive codes, which are not MDS. More precisely, for an integer <math><msub><mrow><mi>m</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>≥</mo><mn>2</mn></math> and a prime power q, we provide a method to construct optimal and almost optimal <math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math>-additive LRCs over <math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><msub><mrow><mi>m</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></msup></mrow></msub></math> with locality r that relies on the existence of certain special polynomials over <math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math>, which we shall refer to as <math><mo>(</mo><mi>r</mi><mo>,</mo><msub><mrow><mi>m</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>)</mo></math>-good polynomials over <math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math>, (note that <math><mo>(</mo><mi>r</mi><mo>,</mo><msub><mrow><mi>m</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>)</mo></math>-good polynomials over <math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math> coincide with r-good polynomials over <math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math> when <math><msub><mrow><mi>m</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><mn>1</mn></math>). We also derive sufficient conditions under which <math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math>-additive LRCs over <math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><msub><mrow><mi>m</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></msup></mrow></msub></math> constructed using the aforementioned method are optimal. We further provide four general methods to construct <math><mo>(</mo><mi>r</mi><mo>,</mo><msub><mrow><mi>m</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>)</mo></math>-good polynomials over <math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math>, which give rise to several classes of optimal and almost optimal LRCs over <math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><msub><mrow><mi>m</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></msup></mrow></msub></math> with locality r. To illustrate these results, we list several optimal LRCs over <math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><msub><mrow><m

有限域上的加法码是线性码的自然扩展，在构建量子纠错码时非常有用。在本文中，我们首先研究了有限域上加法 MDS 码的局部性特性，其对偶码也是 MDS。我们进一步提供了一种方法，用属于非 MDS 的加法码系列的新参数来构造最优和几乎最优的 LRC。更确切地说，对于整数 m0≥2 和素数幂 q，我们提供了一种方法来构造 Fqm0 上具有局部性 r 的最优和近似最优 Fq 附加 LRC，这种方法依赖于 Fq 上某些特殊多项式的存在，我们将其称为 Fq 上的（r,m0）-好多项式（注意，当 m0=1 时，Fq 上的（r,m0）-好多项式与 Fq 上的（r-好多项式）重合）。我们还推导出充分条件，在这些条件下，用上述方法构造的 Fq 上的 Fq-additive LRC 是最优的。为了说明这些结果，我们列出了几个带有新参数的 Fqm0 上最优 LRC。最后，我们考虑了 m0=1 的情况，得到了 Fq 上一些新的 r-good 多项式，从而构建了具有局部性 r 的 Fq 上最优线性 LRC。

引用次数: 0

The generalized Suzuki curve 广义铃木曲线

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-10-11 DOI: 10.1016/j.ffa.2024.102514

Saeed Tafazolian

We present a characterization of the generalized Suzuki curve, focusing on its genus and automorphism group.

我们介绍了广义铃木曲线的特征，重点是它的属和自变群。

引用次数: 0

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Finite Fields and Their Applications

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