Finite Fields and Their Applications最新文献

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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

Binary polycyclic codes associated with x2η+1+x2η+1: Hamming distance, duality, reversibility and LCD properties 与x2η+1+x2η+1相关的二进制多循环码：汉明距离、对偶性、可逆性和LCD性质

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-10-17 DOI: 10.1016/j.ffa.2025.102741

Sujata Bansal, Pramod Kumar Kewat

This work explores binary polycyclic codes associated with the polynomial

x^{2^{η + 1}} + x^{2^{η}} + 1

, which is the

2^{η}

-th power of

x^{2} + x + 1

for every integer

η \geq 1

. We provide an in-depth structural analysis of these codes and compute the exact Hamming distance of each of these binary polycyclic codes. Furthermore, we determine the parity-check matrices and examine the Euclidean duals and annihilator duals for these polycyclic codes. Our analysis reveals that these codes are reversible and, in certain cases, are Linear Complementary Dual (LCD) codes. This discovery highlights the potential of these codes in practical applications such as communication systems, data storage, consumer electronics, and cryptography. We also propose a conjecture that suggests all such polycyclic codes can be LCD.

本文研究了与多项式x2η+1+x2η+1相关的二进制多环码，对于每个η≥1的整数，它是x2+x+1的2η-次幂。我们对这些码进行了深入的结构分析，并计算了每个二进制多环码的精确汉明距离。进一步，我们确定了这些多环码的奇偶校验矩阵，并检验了它们的欧几里得对偶和湮灭对偶。我们的分析表明，这些代码是可逆的，在某些情况下，是线性互补双（LCD）代码。这一发现突出了这些代码在通信系统、数据存储、消费电子和密码学等实际应用中的潜力。我们还提出了一个猜想，表明所有这些多循环码都可以是LCD。

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引用次数: 0

Irreducible factorizations of polynomials xpk+1−bx+b over a finite field 有限域上多项式xpk+1 - bx+b的不可约分解

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-10-16 DOI: 10.1016/j.ffa.2025.102740

Xue Jia , Fengwei Li , Huan Sun , Qin Yue

In this paper, we investigate polynomials of the form

f (x) = x^{p^{k} + 1} - b x + b

, where

0 \neq b \in F_{p^{n}}

, p is a prime, and k divides n. By introducing a new approach based on the projective general linear group, we show that the number of zeros of

f (x)

F_{p^{n}}

belongs to

{0, 1, 2, p^{k} + 1}

, and provide explicit criteria on b for each case. We also count the number of such polynomials corresponding to each possible number of zeros. Moreover, for the cases where

f (x)

has at least one zero, we determine its complete irreducible factorization over

F_{p^{n}}

本文研究了形式为f(x)=xpk+1 - bx+b的多项式，其中0≠b∈Fpn， p是素数，k除n。通过引入一种基于射影一般线性群的新方法，证明了f(x)在Fpn中的零个数属于{0,1,2,pk+1}，并给出了每种情况下b的显式判据。我们还计算对应于每个可能的零数的多项式的个数。此外，对于f(x)至少有一个零的情况，我们确定了它在Fpn上的完全不可约分解。

引用次数: 0

An unusual family of supersingular curves of genus five in characteristic two 特征二的五属超奇异曲线的一个不寻常的族

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-10-10 DOI: 10.1016/j.ffa.2025.102736

Dušan Dragutinović

We construct a family of smooth supersingular curves of genus 5 in characteristic 2 with several notable features: its dimension matches the expected dimension of any component of the supersingular locus in genus 5, its members are non-hyperelliptic curves with non-trivial automorphism groups, and each curve in the family admits a double cover structure over both an elliptic curve and a genus-2 curve. We also provide an explicit parametrization of this family.

构造了特征2上的5属光滑超奇异曲线族，它的维数与5属超奇异轨迹的任意分量的期望维数相匹配，它的成员是具有非平凡自同构群的非超椭圆曲线，族中的每条曲线都在椭圆曲线和2属曲线上具有双覆盖结构。我们还提供了这个家族的显式参数化。

引用次数: 0

On a class of complete permutation quadrinomials 关于一类完全置换四项

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-10-10 DOI: 10.1016/j.ffa.2025.102734

Chin Hei Chan , Zhiguo Ding , Nian Li , Xi Xie , Maosheng Xiong , Michael E. Zieve

Let

f (x) = a x^{3 q} + b x^{2 q + 1} + c x^{q + 2} + d x^{3} \in F_{q^{2}} [x]

, where

F_{q^{2}}

is the finite field of order

q^{2}

and

q = 2^{m}

for some positive integer m. Tu et al. (Finite Fields Appl. 68: 1-20, 2020) proposed a sufficient condition under which

f (x)

is a complete permutation on

F_{q^{2}}

. In this paper, we show that this sufficient condition is also necessary, and when

f (x)

is a complete permutation, then

f (x)

and

f (x) + x

are simultaneously linear equivalent to

x^{2} \overline{x}

and

x^{2} \overline{x} + γ x

for some

γ \in F_{q^{2}}^{⁎}

satisfying

ord (γ^{q - 1}) = 3

. This result leads to a complete characterization of the complete permutation quadrinomials of the above form

f (x)

设f(x)=ax3q+bx2q+1+cxq+2+dx3∈Fq2[x]，其中Fq2是q2阶的有限域，对于某正整数m q=2m。Tu等（finite Fields, 68: 1- 20,2020）提出了f(x)是Fq2上的完全置换的充分条件。在本文中，我们证明了这个充分条件也是必要的，并且当f(x)是一个完全置换时，那么对于某些γ∈Fq2满足ord(γq−1)=3,f(x)和f(x)+x同时线性等价于x2x和x2x⊥+γx。这个结果导致了上述形式f(x)的完全置换四项的完全表征。

引用次数: 0

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Finite Fields and Their Applications

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