Finite Fields and Their Applications最新文献

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Counterexamples, constructions, and nonexistence results for optimal ternary cyclic codes 最优三元循环码的反例、构造和不存在性结果

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2026-01-19 DOI: 10.1016/j.ffa.2026.102795

Jingjun Bao , Hanlin Zou

Cyclic codes are an important subclass of linear codes with wide applications in communication systems and data storage systems. In 2013, Ding and Helleseth presented nine open problems on optimal ternary cyclic codes

C_{(1, e)}

. While the first two and the sixth problems have been fully solved, others remain open. In this paper, we advance the study of the third and fourth open problems by providing the first counterexamples to both and constructing two families of optimal codes under certain conditions, thereby partially solving the third problem. Furthermore, we investigate the cyclic codes

C_{(1, e)}

where

e (3^{h} \pm 1) \equiv \frac{3^{m} - a}{2} (\mod 3^{m} - 1)

and a is odd. For

a \equiv 3 (\mod 4)

, we present two new families of optimal codes with parameters

[3^{m} - 1, 3^{m} - 1 - 2 m, 4]

, generalizing known constructions. For

a \equiv 1 (\mod 4)

, we obtain several nonexistence results on optimal codes

C_{(1, e)}

with the aforementioned parameters revealing the constraints of such codes.

循环码是线性码的一个重要子类，在通信系统和数据存储系统中有着广泛的应用。2013年，Ding和Helleseth提出了9个关于最优三元循环码C（1,e）的开放问题。虽然前两个和第六个问题已经完全解决，但其他问题仍然存在。在本文中，我们通过提供第三和第四开放问题的第一个反例，并在一定条件下构造两族最优码，从而部分解决了第三问题。进一步，我们研究了循环码C(1,e)，其中e(3h±1)≡3m−a2（mod3m−1）且a是奇数。对于a≡3(mod4)，我们提出了两个新的最优码族，参数为[3m−1,3m−1−2m,4]，推广了已知结构。对于a≡1(mod4)，我们得到了若干最优码C（1,e）的不存在性结果，用上述参数揭示了这类码的约束条件。

引用次数: 0

Permutation polynomials of index q + 1 over Fq2 指标q的置换多项式 + 1 / Fq2

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2026-01-19 DOI: 10.1016/j.ffa.2026.102796

Xiutao Feng , Qiang Wang

We provide a generic construction of permutation polynomials over

F_{q^{2}}

with index

q + 1

from any permutation polynomial of

F_{q}

. We also extend our construction using polynomials with coefficients in

F_{q^{2}}

such that they are injective over a subset of

F_{q^{2}}^{⁎}

, which corresponds to the set

μ_{q + 1}

of all

(q + 1)

-th roots of unity.

从Fq的任意排列多项式出发，给出了Fq2上索引为q+1的排列多项式的一般构造。我们还使用系数在Fq2中的多项式扩展了我们的构造，使得它们在Fq2的一个子集上内射，该子集对应于所有(q+1)-单位根的集合μq+1。

引用次数: 0

The unit f-sequence for primitive-based f 基于基元的f的单位f序列

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2026-01-19 DOI: 10.1016/j.ffa.2026.102793

Owen J. Brison , J. Eurico Nogueira

We study the behaviour with respect to the operation of lifting of the unit g-sequence, where

g (t)

is a primitive polynomial. We also study the unit f-sequence for primitive-based

f (t)

and show it is closely related to the lifted unit g-sequence where

f (t)

is based on

g (t)

研究了单位g序列关于提升运算的性质，其中g(t)是一个原始多项式。我们还研究了基于基元的f(t)的单位f序列，并证明它与提升的单位g序列密切相关，其中f(t)基于g(t)。

引用次数: 0

Binary linear codes with at most three weights from cyclotomic mappings 从环切分映射得到的最多三个权值的二进制线性码

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2026-01-19 DOI: 10.1016/j.ffa.2026.102798

Heming Cui , Xubo Zhao , Qiang Wang , Xiaoping Li , Tongjiang Yan

Linear codes with few weights have attracted significant interest due to their wide-ranging applications in secret sharing, authentication codes, association schemes, strongly regular graph, and some other fields. This paper focuses on unifying several existing construction methods for few-weight linear codes, extending the works of Wang et al. (2015) [24], Wu et al. (2019) [25], and Fang et al. (2023) [10]. In our code construction, we introduce a novel index set

J

, whose cardinality and structural properties are shown to critically influence both the length and weight distribution of the resulting few-weight linear codes. By employing cyclotomic mappings and choosing the more general defining sets, several new classes of binary linear codes with at most three weights are constructed. Our framework subsumes all aforementioned constructions as special cases and enlarges the spectrum of attainable parameters. The weight distributions of the corresponding linear codes are also explicitly determined. We also demonstrate that some of the linear codes constructed in this paper are optimal in the sense that they have the best known parameters in the tables maintained by Markus Grassl and/or optimal in the sense that they meet certain bounds on linear codes.

低权重线性码由于其在秘密共享、认证码、关联方案、强正则图等领域的广泛应用而引起了人们的广泛关注。本文的重点是统一几种现有的小权重线性码的构建方法，扩展了Wang等人（2015）[24]，Wu等人（2019）[25]和Fang等人（2023）[10]的工作。在我们的代码构造中，我们引入了一个新的索引集J，它的基数和结构性质被证明对得到的少权重线性代码的长度和权重分布都有重要影响。通过采用环切映射和选择更一般的定义集，构造了几种新的最多三个权重的二元线性码。我们的框架将所有上述结构作为特殊情况纳入，并扩大了可获得参数的范围。明确地确定了相应线性码的权值分布。我们还证明了本文构造的一些线性码是最优的，因为它们在Markus Grassl维护的表中具有最知名的参数，或者在满足线性码的某些界的意义上是最优的。

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引用次数: 0

Hermitian-Singer functional and differential codes 埃尔米特-辛格功能码和微分码

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2026-01-16 DOI: 10.1016/j.ffa.2026.102794

Gábor Korchmáros , Federico Romaniello , Valentino Smaldore

Algebraic geometry codes on the Hermitian curve have been the subject of several papers, since they happen to have good performances and large automorphism groups. Here, those arising from the Singer cycle of the Hermitian curve are investigated.

厄米曲线上的代数几何码已经成为一些论文的主题，因为它们恰好具有良好的性能和大的自同构群。本文研究了由厄米曲线的辛格周期引起的问题。

引用次数: 0

De Bruijn tori without zeros: a field-theoretic perspective 没有零的德布鲁因托里：场理论视角

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2026-01-15 DOI: 10.1016/j.ffa.2026.102790

Ming Hsuan Kang, Yu Hsuan Hsieh

We present an algebraic construction of trace-based De Bruijn tori over finite fields, focusing on the nonzero variant that omits the all-zero pattern. The construction arranges nonzero field elements on a toroidal grid using two multiplicatively independent generators, with values obtained by applying a fixed linear map, typically the field trace.

We characterize sampling patterns as subsets whose associated field elements form an

F_{p}

-basis, and show that column structures correspond to cyclic shifts of De Bruijn sequences determined by irreducible polynomials over subfields. Recursive update rules based on multiplicative translations enable efficient computation.

我们给出了有限域上基于迹的德布鲁因托里的代数构造，重点讨论了忽略全零模式的非零变体。该构造使用两个相乘独立的生成器在环面网格上排列非零场元素，其值通过应用固定的线性映射（通常是场迹）获得。我们将采样模式描述为子集，其相关的域元素形成一个fp基，并表明列结构对应于子域上由不可约多项式决定的De Bruijn序列的循环移位。基于乘法转换的递归更新规则实现了高效的计算。

引用次数: 0

Quasi-optimum distance flag codes 准最佳距离标志码

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2026-01-15 DOI: 10.1016/j.ffa.2026.102799

Clementa Alonso-González, Miguel Ángel Navarro-Pérez

A flag is a sequence of nested subspaces of a given ambient space

F_{q}^{n}

over a finite field

F_{q}

. In network coding, a flag code is a set of flags, all of them with the same sequence of dimensions, the type vector. In this paper, we investigate quasi-optimum distance flag codes, i.e., those attaining the second best possible distance value. We characterize them and present upper bounds for their cardinality. Moreover, we propose a systematic construction for every choice of the type vector by using partial spreads and sunflowers. For flag codes with lower minimum distance, we adapt the previous construction and provide some results towards their characterization, especially in the case of the third best possible distance value.

标志是给定环境空间Fqn在有限域Fq上的嵌套子空间序列。在网络编码中，标志码是一组标志，它们都具有相同的维数序列，即类型向量。在本文中，我们研究了准最优距离标志码，即那些达到次优可能距离值的标志码。我们对它们进行了表征，并给出了它们的基数的上界。此外，我们提出了一个系统的结构，为每一个选择的类型向量使用部分蔓延和向日葵。对于最小距离较低的标志码，我们调整了之前的结构，并对其表征提供了一些结果，特别是在第三最佳可能距离值的情况下。

引用次数: 0

Permutation polynomials of the form (xq−x+δ)i(q−1)+1+L(x) over Fq2 形式为（xq−x+δ）i(q−1)+1+L(x) / Fq2的置换多项式

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2026-01-14 DOI: 10.1016/j.ffa.2026.102797

Rohit Gupta , Amritanshu Rai

Let q be a power of a prime number and let

F_{q}

be the finite field with q elements. Let

δ \in F_{q^{2}}

be arbitrary. In this paper, we give a relationship between the permutation property of polynomials over

F_{q^{2}}

of the forms

g (x^{q} - x + δ) + c x^{q} + d x

and

g {(x)}^{q} - g (x) + c x^{q} + d x

where

c, d \in F_{q}^{⁎}

g (x) \in F_{q^{2}} [x]

. Further, we find the necessary and sufficient conditions on the coefficients c and d such that polynomials of the forms

{(x^{q} - x + δ)}^{i (q - 1) + 1} + c x

and

{(x^{q} - x + δ)}^{i (q - 1) + 1} + c x^{q} + d x

permute

F_{q^{2}}

. Moreover, some results of this article supersede certain results in the related literature.

设q是质数的幂，设Fq是有q个元素的有限域。设δ∈Fq2是任意的。本文给出了形式为g(xq−x+δ)+cxq+dx和g(x)q−g(x)+cxq+dx的多项式在Fq2上的置换性质之间的关系，其中c，d∈Fq, g(x)∈Fq2[x]。进一步，我们找到了系数c和d的充要条件，使得多项式的形式为（xq−x+δ）i(q−x+δ) +1+cx和（xq−x+δ）i(q−1)+1+cxq+dx可以置换Fq2。此外，本文的一些结果取代了相关文献中的某些结果。

引用次数: 0

Function-correcting codes with homogeneous distance 齐次距离函数校正码

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2026-01-13 DOI: 10.1016/j.ffa.2026.102791

Huiying Liu, Hongwei Liu

Function-correcting codes are designed to reduce redundancy of codes when protecting function values of information against errors. As generalizations of Hamming weights and Lee weights over

Z_{4}

, homogeneous weights are used in codes over finite rings. In this paper, we introduce function-correcting codes with homogeneous distance denoted by FCCHDs, which extend function-correcting codes with Hamming distance. We first define D-homogeneous distance codes. We use D-homogeneous distance codes to characterize connections between the optimal redundancy of FCCHDs and lengths of these codes for some matrices D. By these connections, we obtain several bounds of the optimal redundancy of FCCHDs for some functions. In addition, we also construct FCCHDs for homogeneous weight functions and homogeneous weight distribution functions. Specially, redundancies of some codes we construct in this paper reach the optimal redundancy bounds.

功能校正码的设计是为了在保护信息的功能值不受错误影响时减少代码的冗余。作为Z4上的Hamming权和Lee权的推广，齐次权用于有限环上的码。本文引入了用FCCHDs表示的具有齐次距离的功能纠错码，它扩展了具有汉明距离的功能纠错码。首先定义d齐次距离码。我们用d -齐次距离码来描述一些矩阵d的fcchd的最优冗余度和这些码的长度之间的联系，通过这些联系，我们得到了一些函数的fcchd的最优冗余度的几个界。此外，我们还构造了齐次权函数和齐次权分布函数的FCCHDs。特别地，本文构造的一些码的冗余达到了最优冗余界。

引用次数: 0

Twisted group algebra of dihedral groups over finite fields 有限域上二面体群的扭曲群代数

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2026-01-13 DOI: 10.1016/j.ffa.2026.102792

André Duarte

Let

F_{q}

be a finite field and

D_{n}

is the dihedral group of order 2n. We present formulas for a complete set of 2-cocycles of

D_{n}

over

F_{q}

and compute the primitive central idempotents of

F_{q}^{α} D_{n}

. We conclude by describing the Wedderburn decomposition of

F_{q}^{α} D_{n}

and the irreducible projective representations of

D_{n}

over

F_{q}

设Fq是一个有限域，Dn是2n阶的二面体群。我们给出了Dn / Fq的2环完备集的公式，并计算了FqαDn的原始中心幂等。最后，我们描述了FqαDn的Wedderburn分解和Dn / Fq的不可约投影表示。

引用次数: 0

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