Finite Fields and Their Applications最新文献

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On quadratic character sums over quartics 关于四分位数上的二次字符和

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-11-03 DOI: 10.1016/j.ffa.2025.102755

Bogdan Nica

We obtain transformation formulas for quadratic character sums with quartic and cubic polynomial arguments.

得到了具有四次和三次多项式参数的二次特征和的变换公式。

引用次数: 0

Continued fractions and indefinite binary quadratic forms over Fq[t] Fq上的连分式和不定二元二次型[t]

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-10-31 DOI: 10.1016/j.ffa.2025.102747

Jorge Morales

The relation between cycles of indefinite binary quadratic forms over

Z

and continued fractions is classical and well-known. We describe a similar relation for binary quadratic forms over the polynomial ring

F_{q} [t]

, where q is a power of an odd prime. In this context, the cycles of the classical theory are replaced by orbits of the metacyclic group

F_{q}^{⁎} ⋊ Z

acting on the set of reduced forms of a given discriminant, where each orbit corresponds to a proper equivalence class.

Z上不定二元二次型的循环与连分式之间的关系是经典而众所周知的。我们描述了多项式环Fq[t]上二元二次型的类似关系，其中q是奇素数的幂。在这种情况下，经典理论的循环被作用于给定判别式的约简形式集合上的亚环群Fq Z的轨道所取代，其中每个轨道对应于一个适当的等价类。

引用次数: 0

The Erdős-Rado sunflower problem for vector spaces 向量空间的Erdős-Rado向日葵问题

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-10-30 DOI: 10.1016/j.ffa.2025.102746

Ferdinand Ihringer , Andrey Kupavskii

The famous Erdős-Rado sunflower conjecture suggests that an s-sunflower-free family of k-element sets has size at most

{(C s)}^{k}

for some absolute constant C. In this note, we investigate the analog problem for k-spaces over the field with q elements. For

s \geq k + 1

, we show that the largest s-sunflower-free family

F

satisfies

1 \leq | F | / q^{(s - 1) (\begin{matrix} k + 1 \\ 2 \end{matrix}) - k} \leq {(q / (q - 1))}^{k} .

For

s \leq k

, we show that

q^{- (\begin{matrix} k + 1 \\ 2 \end{matrix})} \leq | F | / q^{(s - 1) (\begin{matrix} k + 1 \\ 2 \end{matrix}) - k} \leq {(q / (q - 1))}^{k} .

Our lower bounds rely on an iterative construction that uses lifted maximum rank-distance (MRD) codes.

著名的Erdős-Rado向日葵猜想表明，对于某个绝对常数c，一个无s-向日葵的k元素集合族的大小最多为（Cs）k。在本文中，我们研究了具有q个元素的域上k空间的模拟问题。当s≥k+1时，我们证明了最大的s-无向日葵族F满足1≤|F|/q(s−1)(k+12)−k≤(q/(q−1))k。对于s≤k,我们展示thatq−(k + 12)≤F | | / q (s−1)k (k + 12)−≤(q /(问−1))k。我们的下界依赖于使用提升最大秩距离（MRD）代码的迭代构造。

引用次数: 0

The third generalized covering radius for binary primitive double-error-correcting BCH codes 二元基元双纠错BCH码的第三种广义覆盖半径

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-10-30 DOI: 10.1016/j.ffa.2025.102749

Ferruh Özbudak, İlknur Öztürk

We prove that the third generalized covering radius of binary primitive double-error-correcting BCH codes of length

2^{m} - 1

is 7 if

m \geq 8

is an even integer. We also prove that the third generalized covering radius of binary primitive double-error-correcting BCH codes of length

2^{m} - 1

is either 6 or 7 if

m \geq 9

is an odd integer. We use some methods derived from the theory of algebraic curves over finite fields in our proofs and we obtain some further related results.

证明了当m≥8为偶数时，长度为2m−1的二元基元双纠错BCH码的第三广义覆盖半径为7。证明了当m≥9为奇数时，长度为2m−1的二元基元双纠错BCH码的第三广义覆盖半径为6或7。我们利用有限域上代数曲线理论中导出的一些方法进行了证明，并得到了一些进一步的相关结果。

引用次数: 0

Cardinality-consistent flag codes with larger cardinality 具有较大基数的基数一致的标志代码

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-10-29 DOI: 10.1016/j.ffa.2025.102750

Junfeng Jia, Yanxun Chang

Flag codes, as a generalization of subspace codes, can transmit more information since the subspace channel is used many times. In this paper, we construct optimum distance flag codes of the (generalized) full admissible type

t = (1, \dots, k, n - k, \dots, n - 1)

F_{q}^{n}

with cardinality

\sum_{i = 1}^{s - 1} q^{i k + h} + 1

, where

n = s k + h

with

s \geq 2

and

0 \leq h < k

. Let

A_{q}^{f} (n, D^{(t, n)}, t)

denote the maximum cardinality of such codes. We provide a lower bound for this quantity. We further present a systematic construction of cardinality-consistent flag codes with larger cardinality for general flag distances. By the composition of subspace polynomials, we construct cardinality-consistent cyclic flag codes on

F_{q^{n}}

with larger cardinality than those presented in the literature.

标志码作为子空间码的泛化，由于子空间信道被多次使用，可以传输更多的信息。本文在基数∑i=1s - 1qik+h+1的Fqn上构造了（广义）完全可容许型t=（1，…，k,n - k，…，n - 1）的最优距离标志码，其中n=sk+h, s≥2,0≤h<k。设Aqf（n，D(t,n),t）表示这些码的最大基数。我们给出了这个量的下界。我们进一步提出了对一般旗距具有较大基数的基数一致旗码的系统构造。通过子空间多项式的组合，我们在Fqn上构造了基数一致的循环标志码，其基数大于已有的循环标志码。

引用次数: 0

Normal and primitive normal elements with prescribed traces in intermediate extensions of finite fields 有限域中间扩展中具有规定迹的正规元和原始正规元

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-10-27 DOI: 10.1016/j.ffa.2025.102745

Arpan Chandra Mazumder , Giorgos Kapetanakis , Dhiren Kumar Basnet

In this article, we study the existence and distribution of elements in finite field extensions with prescribed traces in several intermediate extensions that are also either normal or primitive normal. In the former case, we fully characterize the conditions under which such elements exist and provide an explicit enumeration of these elements. In the latter case we provide asymptotic results.

本文研究了具有规定迹的有限域扩展中元素的存在性和分布。在前一种情况下，我们充分描述了这些元素存在的条件，并提供了这些元素的显式枚举。在后一种情况下，我们提供渐近结果。

引用次数: 0

Some permutation pentanomials over finite fields of even characteristic 偶特征有限域上的一些置换五反常项

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-10-23 DOI: 10.1016/j.ffa.2025.102742

Farhana Kousar , Maosheng Xiong

In a recent paper [30] Zhang et al. constructed 17 families of permutation pentanomials of the form

x^{t} + x^{r_{1} (q - 1) + t} + x^{r_{2} (q - 1) + t} + x^{r_{3} (q - 1) + t} + x^{r_{4} (q - 1) + t}

over

F_{q^{2}}

where

q = 2^{m}

. In this paper for 14 of these 17 families we provide a simple explanation as to why they are permutations. We also extend these 14 families into three general classes of permutation pentanomials over

F_{q^{2}}

在最近的一篇论文[30]中，Zhang等人构造了17个形式为xt+xr1(q−1)+t+xr2(q−1)+t+xr3(q−1)+t+xr4(q−1)+t / Fq2的置换五反常族，其中q=2m。在这篇论文中，我们对这17个家族中的14个提供了一个简单的解释，为什么它们是排列。我们还将这14个科扩展到Fq2上的3类置换五反常。

引用次数: 0

Finite fields whose members are the sum of a potent and a 4-potent 有限域，其成员是幂次域和4幂次域的和

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-10-22 DOI: 10.1016/j.ffa.2025.102739

Stephen D. Cohen , Peter V. Danchev , Tomás Oliveira e Silva

We classify those finite fields

F_{q}

whose members are the sum of an n-potent element with

n > 1

and a 4-potent element. It is shown that there are precisely ten non-trivial pairs

(q, n)

for which this is the case. This continues a recent publication by Abyzov et al. (2024) [1] in which the tripotent version was examined in-depth, inasmuch as it extends recent results in this seam of research established by Abyzov and Tapkin (2024) [4].

我们对有限域Fq进行了分类，这些域的成员是一个n强元素和一个4强元素的和。结果表明，恰好有十个非平凡对（q,n）存在这种情况。这是Abyzov等人（2024）[1]最近发表的一篇文章的延续，其中对三能性版本进行了深入研究，因为它扩展了Abyzov和Tapkin(2024)[1]建立的这一研究领域的最新成果。

引用次数: 0

On sum-free functions 关于无和函数

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-10-21 DOI: 10.1016/j.ffa.2025.102744

Alyssa Ebeling , Xiang-dong Hou , Ashley Rydell , Shujun Zhao

A function from

F_{2^{n}}

F_{2^{n}}

is said to be kth order sum-free if the sum of its values over each k-dimensional

F_{2}

-affine subspace of

F_{2^{n}}

is nonzero. This notion was recently introduced by C. Carlet as, among other things, a generalization of APN functions. At the center of this new topic is a conjecture about the sum-freedom of the multiplicative inverse function

f_{inv} (x) = x^{- 1}

(with

0^{- 1}

defined to be 0). It is known that

f_{inv}

is 2nd order (equivalently,

(n - 2)

th order) sum-free if and only if n is odd, and it is conjectured that for

3 \leq k \leq n - 3

f_{inv}

is never kth order sum-free. The conjecture has been confirmed for even n but remains open for odd n. In the present paper, we show that the conjecture holds under each of the following conditions: (1)

n = 13

; (2)

3 | n

; (3)

5 | n

; (4) the smallest prime divisor l of n satisfies

(l - 1) (l + 2) \leq (n + 1) / 2

. We also determine the “right” q-ary generalization of the binary multiplicative inverse function

f_{inv}

in the context of sum-freedom. This q-ary generalization not only maintains most results for its binary version, but also exhibits some extraordinary phenomena that are not observed in the binary case.

如果一个从F2n到F2n的函数在F2n的每一个k维的f2仿射子空间上的值的和是非零的，那么这个函数就是无k阶和的。这个概念是最近由C. Carlet引入的，作为APN函数的推广。这个新主题的中心是一个关于乘法反函数finv(x)=x−1（其中0−1定义为0）的和自由度的猜想。已知finv是二阶（即（n−2）阶）自由和当且仅当n为奇数，并且推测当3≤k≤n−3时，finv绝不是第k阶自由和。对于偶数n，该猜想已被证实，但对于奇数n，该猜想仍然是开放的。在本文中，我们证明了该猜想在下列条件下成立：(1)n=13；(2) 3 | n;(3) 5 | n;(4) n的最小素数因子l满足（l−1）(l+2)≤(n+1)/2。我们还确定了二元乘法反函数finv在自由和情况下的“正确”q-ary泛化。这种q-ary泛化不仅保留了其二进制版本的大多数结果，而且还展示了一些在二进制情况下没有观察到的特殊现象。

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引用次数: 0

Further results on permutation pentanomials over Fq3 in characteristic two 特征二上Fq3上排列五反常的进一步结果

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-10-20 DOI: 10.1016/j.ffa.2025.102743

Tongliang Zhang , Lijing Zheng , Hengtai Wang , Jie Peng , Yanjun Li

Let

q = 2^{m}

. In a recent paper [34], Zhang and Zheng investigated several classes of permutation pentanomials of the form

ϵ_{0} x^{d_{0}} + L (ϵ_{1} x^{d_{1}} + ϵ_{2} x^{d_{2}})

over

F_{q^{3}} (d_{0} = 1, 2, 4)

with a certain linearized polynomial

L (x)

. They applied the multivariate method and specific techniques to analyze the number of solutions of certain equations, and proposed an open problem: the permutation property of some pentanomials of this form remains unproven. In this paper, inspired by the idea of [12], we further characterize the permutation property of such pentanomials over

F_{q^{3}} (d_{0} = 1, 2, 4)

. The techniques presented in this paper will be useful for investigating more new classes of permutation polynomials.

让q = 2 m。在最近的一篇论文[34]中，Zhang和Zheng研究了几种形式为ϵ0xd0+L（ϵ1xd1+ϵ2xd2） / Fq3（d0=1,2,4）的具有一定线性化多项式L(x)的置换五反常。他们运用多元方法和特定技术分析了某些方程的解的个数，并提出了一个开放性问题：一些这种形式的五反常项的置换性质尚未得到证明。在本文中，受[12]思想的启发，我们进一步刻画了Fq3（d0=1,2,4）上这类五反常的置换性质。本文提出的技术将有助于研究更多新的置换多项式类。

引用次数: 0

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