Finite Fields and Their Applications最新文献

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Corrigendum to “Higher Grassmann codes II” [Finite Fields Appl. 89 (2023) 102211] “高等格拉斯曼规范II”的勘误表[有限域应用89 (2023)102211]

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2026-01-01 Epub Date: 2025-07-17 DOI: 10.1016/j.ffa.2025.102700

Mahir Bilen Can , Roy Joshua , G.V. Ravindra

引用次数: 0

New constructions of permutation polynomials of the form x+γTrqq3(h(x)) over finite fields with even characteristic 偶特征有限域上形式为x+γTrqq3（h(x)）的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 : 2026-01-01 Epub Date: 2025-07-09 DOI: 10.1016/j.ffa.2025.102694

Sha Jiang, Kangquan Li, Longjiang Qu

Permutation polynomials over finite fields have applications in many areas of mathematics and engineering. Particularly, permutation polynomials of the form

x + γ {Tr}_{q}^{q^{n}} (h (x))

have been studied for a long time. In this paper, we further investigate permutation polynomials of the form

x + γ {Tr}_{q}^{q^{3}} (h (x))

over finite fields with even characteristic. For one thing, by choosing functions

h (x)

with a low q-degree, we propose four classes of permutation polynomials of the form

x + γ {Tr}_{q}^{q^{3}} (h (x))

over

F_{q^{3}}

. For the other thing, we give seven classes of permutations of the form

x + {Tr}_{q}^{q^{3}} (h (x))

with binomials

h (x)

over

F_{q^{3}}

. Finally, we also show that the permutation polynomials constructed in this paper are not quasi-multiplicative equivalent to the known permutation polynomials.

有限域上的置换多项式在数学和工程的许多领域都有应用。特别是形式为x+γTrqqn（h(x)）的置换多项式已经被研究了很长时间。本文进一步研究了具有偶特征的有限域上形式为x+γTrqq3（h(x)）的置换多项式。一方面，通过选择低q度的函数h(x)，我们提出了四类形式为x+γTrqq3(h(x)) / Fq3的置换多项式。另一方面，我们给出了7种形式为x+Trqq3（h(x)）的二项式h(x) / Fq3的排列。最后，我们还证明了本文构造的置换多项式与已知的置换多项式不是拟乘法等价的。

引用次数: 0

Generator polynomial matrices of the Galois hulls of multi-twisted codes 多扭曲码伽罗瓦壳的生成多项式矩阵

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2026-01-01 Epub Date: 2025-08-04 DOI: 10.1016/j.ffa.2025.102712

Ramy Taki Eldin , Patrick Solé

In this study, we consider the Euclidean and Galois hulls of multi-twisted (MT) codes over a finite field

F_{p^{e}}

of characteristic p. Let G be a generator polynomial matrix (GPM) of an MT code

C

. For any

0 \leq κ < e

, the κ-Galois hull of

C

, denoted by

h_{κ} (C)

, is the intersection of

C

with its κ-Galois dual. The main result in this paper is that a GPM for

h_{κ} (C)

has been obtained from G. We start by associating a linear code

Q_{G}

with G. We show that

Q_{G}

is quasi-cyclic. In addition, we prove that the dimension of

h_{κ} (C)

is the difference between the dimension of

C

and that of

Q_{G}

. Thus the determinantal divisors are used to derive a formula for the dimension of

h_{κ} (C)

. Finally, we deduce a GPM formula for

h_{κ} (C)

. In particular, we handle the cases of κ-Galois self-orthogonal and linear complementary dual MT codes; we establish equivalent conditions that characterize these cases. Equivalent results can be deduced immediately for the classes of cyclic, constacyclic, quasi-cyclic, generalized quasi-cyclic, and quasi-twisted codes, because they are all special cases of MT codes. Some numerical examples, containing codes with the best-known parameters, are used to illustrate the theoretical results.

本文考虑特征为p的有限域Fpe上的多扭曲码的欧几里得壳和伽罗瓦壳。设G为多扭曲码C的生成多项式矩阵（GPM）。对于任意0≤κ<；e， C的κ-伽罗瓦壳表示为C与其κ-伽罗瓦对偶的交。本文的主要结果是从g中得到了hκ(C)的GPM。我们首先将线性码QG与g关联，并证明了QG是拟循环的。此外，我们还证明了hκ(C)的维数是C与QG的维数之差。因此，行列式除数被用来推导出hκ(C)维数的公式。最后，我们推导出hκ(C)的GPM公式。特别地，我们处理了κ-伽罗瓦自正交和线性互补对偶MT码的情况；我们建立了表征这些情况的等价条件。对于循环码、恒循环码、拟循环码、广义拟循环码和拟扭曲码，由于它们都是MT码的特殊情况，所以可以立即推导出等价的结果。一些数值例子，包含代码与最知名的参数，用来说明理论结果。

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引用次数: 0

Construction of (n,n)-functions with low differential-linear uniformity 低微分线性均匀性（n,n）函数的构造

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2026-01-01 Epub Date: 2025-07-31 DOI: 10.1016/j.ffa.2025.102710

Xi Xie , Nian Li , Qiang Wang , Xiangyong Zeng , Yinglong Du

The differential-linear connectivity table (DLCT), introduced by Bar-On et al. at EUROCRYPT'19, is a novel tool that captures the dependency between the two subciphers involved in differential-linear attacks. This paper is devoted to exploring the differential-linear properties of

(n, n)

-functions. First, by refining specific exponential sums, we propose two classes of power functions over

F_{2^{n}}

with low differential-linear uniformity (DLU). Next, we further investigate the differential-linear properties of

(n, n)

-functions that are polynomials by utilizing power functions with known DLU. Specifically, by combining a cubic function with quadratic functions, and employing generalized cyclotomic mappings, we construct several classes of

(n, n)

-functions with low DLU, including some that achieve optimal or near-optimal DLU compared to existing results.

Bar-On等人在EUROCRYPT'19上介绍的微分线性连通性表（dct）是一种捕获微分线性攻击中涉及的两个子密码之间依赖关系的新工具。本文研究(n,n)-函数的微分-线性性质。首先，通过细化特定的指数和，我们提出了F2n上具有低微分线性均匀性（DLU）的两类幂函数。接下来，我们利用已知DLU的幂函数进一步研究多项式函数（n,n）的微分线性性质。具体而言，通过将三次函数与二次函数相结合，并采用广义环切面映射，我们构建了几类具有低DLU的(n,n)-函数，包括一些与现有结果相比达到最优或接近最优DLU的函数。

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引用次数: 0

Construction of MDS Euclidean self-dual codes via multiple subsets 基于多子集的MDS欧几里德自对偶码的构造

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2026-01-01 Epub Date: 2025-08-29 DOI: 10.1016/j.ffa.2025.102718

Weirong Meng , Weijun Fang , Fang-Wei Fu , Haiyan Zhou , Ziyi Gu

MDS self-dual codes have good algebraic structure, and their parameters are completely determined by the code length. In recent years, the construction of MDS Euclidean self-dual codes with new lengths has become an important issue in coding theory. In this paper, we are committed to constructing new MDS Euclidean self-dual codes via generalized Reed-Solomon (GRS) codes and their extended (EGRS) codes. The main effort of our constructions is to find suitable subsets of finite fields as the evaluation sets, ensuring that the corresponding (extended) GRS codes are Euclidean self-dual. Firstly, we present a method for selecting evaluation sets from multiple intersecting subsets and provide a theorem to guarantee that the chosen evaluation sets meet the desired criteria. Secondly, based on this theorem, we construct six new classes of MDS Euclidean self-dual codes using the norm function, as well as the union of three multiplicity subgroups and their cosets respectively. Finally, in our constructions, the proportion of possible MDS Euclidean self-dual codes exceeds 85%, which is much higher than previously reported results.

MDS自对偶码具有良好的代数结构，其参数完全由码长决定。近年来，构造具有新长度的MDS欧几里得自对偶码已成为编码理论中的一个重要问题。在本文中，我们致力于通过广义Reed-Solomon （GRS）码及其扩展（EGRS）码构造新的MDS欧几里得自对偶码。我们构造的主要工作是找到合适的有限域子集作为评估集，确保相应的（扩展的）GRS码是欧几里得自对偶的。首先，提出了一种从多个相交子集中选择评价集的方法，并给出了一个保证所选评价集满足期望准则的定理。其次，在此定理的基础上，利用范数函数构造了6类新的MDS欧几里德自对偶码，并分别构造了3个多重子群及其余集的并。最后，在我们的结构中，可能的MDS欧几里得自对偶码的比例超过85%，远远高于先前报道的结果。

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引用次数: 0

Bounds on the size of (r,δ)-locally repairable codes for fixed values q and d （r,δ）-定值q和d的局部可修码的大小界限

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2026-01-01 Epub Date: 2025-07-11 DOI: 10.1016/j.ffa.2025.102697

Wenqin Zhang , Chen Yuan , Yuan Luo , Nian Li

Erasure codes can strengthen fault-tolerance and reliability in distributed storage systems. One of these, locally repairable codes (LRCs), plays a crucial role. A locally repairable code with locality r (called r-LRC) can recover any coded symbol by accessing at most r other coded symbols. The original definition of LRCs is used to repair a single failed node. To address the practical concern of multiple node failures, the concept of LRCs with locality

(r, δ)

was introduced by Prakash et al. (2012) which can be seen as a generalization of r-LRCs. Since then, the bounds and constructions of

(r, δ)

-LRCs have been extensively studied.

This paper is dedicated to both the bounds and constructions of

(r, δ)

-LRCs with disjoint local repair groups over a finite field

F_{q}

, where the parameters r, δ, the alphabet size q, and the minimum distance d are fixed constants, while the code length n tends to infinity. Inspired by the method of the classical Gilbert-Varshamov (GV) bound, we first derive an asymptotic Gilbert-Varshamov-type bound for

(r, δ)

-LRCs in this regime. We manage to show that this GV-type bound works as a threshold for random linear

(r, δ)

-LRCs with disjoint local repair groups using the first and second moment methods. As a corollary, such a random linear

(r, δ)

-LRC has a high probability of attaining the GV-type bound. As an analogue to the classic GV-type bound, we present two constructions of

(r, δ)

-LRCs that each beats this GV-type bound. One construction is obtained from a straightforward concatenation of an outer BCH code and an inner

[r + δ - 1, r]

MDS code. Another construction is based on the Kronecker product of two matrices. To complement our results, the case that n is large but finite is also considered. In this regime, we provide an explicit upper bound for the binary r-LRCs, which is an improvement over the one in Ma and Ge (2019). Furthermore, this bound is shown to be tight for some specific parameters.

Erasure code可以增强分布式存储系统的容错性和可靠性。其中，局部可修复编码（lrc）起着至关重要的作用。局部性为r的局部可修复码（称为r- lrc）可以通过访问最多r个其他编码符号来恢复任何编码符号。lrc的原始定义用于修复单个故障节点。为了解决多节点故障的实际问题，Prakash等人（2012）引入了局域性（r,δ）的lrc概念，可以看作是r- lrc的推广。从那时起，(r,δ)- lrc的界和结构得到了广泛的研究。本文研究有限域Fq上具有不相交局部修群的(r,δ)- lrc的界和构造，其中参数r，δ，字母大小q和最小距离d是固定常数，而编码长度n趋于无穷。在经典Gilbert-Varshamov （GV）界方法的启发下，我们首先导出了(r,δ)- lrc的渐近Gilbert-Varshamov型界。我们使用第一和第二矩方法成功地证明了这种gv型边界作为具有不接合的局部修复群的随机线性(r,δ)- lrc的阈值。作为一个推论，这样的随机线性(r,δ)-LRC有很高的概率达到gv型界。作为经典gv型界的类似物，我们提出了(r,δ)- lrc的两种结构，它们都优于该gv型界。一种结构是由外部BCH码和内部[r+δ−1，r] MDS码的直接连接得到的。另一种构造是基于两个矩阵的克罗内克积。为了补充我们的结果，我们还考虑了n很大但有限的情况。在这种情况下，我们为二元r- lrc提供了明确的上限，这是对Ma和Ge（2019）中的上限的改进。此外，对于某些特定参数，该界是紧的。

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引用次数: 0

Nullities of multisets and value sets of multivariate polynomials 多元多项式的多集和值集的零值

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2026-01-01 Epub Date: 2025-07-23 DOI: 10.1016/j.ffa.2025.102709

Wei Cao

We extend the nullity for a finite 1-tuple multiset, which was introduced by Nica, to a finite m-tuple multiset, and then use it to give an upper bound for the value set of a multivariate polynomial over the multisets drawn from a field. Our results generalize and refine two generalizations of original Wan's upper bound for the value set of a univariate polynomial in finite fields.

本文将Nica提出的有限一元元组多集的零性推广到有限一元元组多集，并利用它给出了由域绘制的多集上的多元多项式的值集的上界。我们的结果推广和改进了有限域中单变量多项式值集的原始Wan上界的两个推广。

引用次数: 0

The number of solutions of diagonal cubic equations over Galois rings GR(p2,r) 伽罗瓦环上对角三次方程的解数

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2026-01-01 Epub Date: 2025-07-29 DOI: 10.1016/j.ffa.2025.102711

Na Chen, Haiyan Zhou

<div><div>Let <math><mi>R</mi><mo>=</mo><mi>G</mi><mi>R</mi><mo>(</mo><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>,</mo><mi>r</mi><mo>)</mo></math> be a Galois ring of characteristic <math><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup></math> with cardinality <math><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn><mi>r</mi></mrow></msup></math>, where p is a prime. Let <math><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>z</mi><mo>)</mo></math>, <math><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>z</mi><mo>)</mo></math>, <math><msubsup><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>′</mo></mrow></msubsup><mo>(</mo><mi>z</mi><mo>)</mo></math> and <math><msubsup><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>′</mo></mrow></msubsup><mo>(</mo><mi>z</mi><mo>)</mo></math> denote the number of solutions of the equations <math><msubsup><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>3</mn></mrow></msubsup><mo>+</mo><mo>…</mo><mo>+</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msubsup><mo>=</mo><mi>z</mi></math>, <math><msubsup><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>3</mn></mrow></msubsup><mo>+</mo><mo>…</mo><mo>+</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msubsup><mo>+</mo><mi>z</mi><msubsup><mrow><mi>x</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mrow><mn>3</mn></mrow></msubsup><mo>=</mo><mn>0</mn></math>, <math><mi>p</mi><msubsup><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></msubsup><mo>+</mo><mi>p</mi><msubsup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>3</mn></mrow></msubsup><mo>+</mo><mo>…</mo><mo>+</mo><mi>p</mi><msubsup><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msubsup><mo>=</mo><mi>z</mi></math> and <math><mi>p</mi><msubsup><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></msubsup><mo>+</mo><mi>p</mi><msubsup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>3</mn></mrow></msubsup><mo>+</mo><mo>…</mo><mo>+</mo><mi>p</mi><msubsup><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msubsup><mo>+</mo><mi>z</mi><msubsup><mrow><mi>x</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mrow><mn>3</mn></mrow></msubsup><mo>=</mo><mn>0</mn></math>, respectively. In this paper, we show that for any <math><mi>z</mi><mo>∈</mo><mi>R</mi></math>, the generating functions <math><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mo>∞</mo></mrow></msub

设R=GR（p2， R）是一个特征为p2，基数为p2r的伽罗瓦环，其中p为素数。令An(z)、Bn(z)、An ' (z)和Bn ' (z)分别表示方程x13+x23+…+xn3=z、x13+x23+…+xn3+zxn+13=0、px13+px23+…+pxn3=z和px13+px23+…+pxn3+zxn+13=0的解个数。本文证明了对于任意z∈R，生成函数∑n=1∞An(z)xn,∑n=1∞Bn(z)xn,∑n=1∞An ' (z)xn和∑n=1∞Bn ' (z)xn是x上的有理函数，并给出了它们的显式表达式。

引用次数: 0

A characterization and an explicit description of all primitive polynomials of degree two 二阶多项式的一个特征和一个明确的描述

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2026-01-01 Epub Date: 2025-08-12 DOI: 10.1016/j.ffa.2025.102716

Gerardo Vega

For polynomials of degree two over finite fields, we present an improvement of Fitzgerald's characterization of primitive polynomials. We then use this new characterization to obtain an explicit, complete, and simple description of all primitive polynomials of degree two over finite fields.

对于有限域上的二阶多项式，我们给出了对Fitzgerald关于原始多项式的描述的改进。然后，我们利用这个新的表征得到了有限域上所有二阶多项式的显式、完整和简单的描述。

引用次数: 0

The distribution of a-numbers of hyperelliptic curves in characteristic three 特征3超椭圆曲线的a数分布

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2026-01-01 Epub Date: 2025-08-21 DOI: 10.1016/j.ffa.2025.102715

Derek Garton , Jeffrey Lin Thunder , Colin Weir

In this paper we present a new approach to counting the proportion of hyperelliptic curves of genus g defined over a finite field

F_{q}

with a given a-number. In characteristic three this method gives exact probabilities for curves of the form

Y^{2} = f (X)

with

f (X) \in F_{q} [X]

monic and cubefree—probabilities that match the data presented by Cais et al. in previous work. These results are sufficient to derive precise estimates (in terms of q) for these probabilities when restricting to squarefree f. As a consequence, for positive integers a and g we show that the nonempty strata of the moduli space of hyperelliptic curves of genus g consisting of those curves with a-number a are of codimension

2 a - 1

. This contrasts with the analogous result for the moduli space of abelian varieties in which the codimensions of the strata are

a (a + 1) / 2

. Finally, our results allow for an alternative heuristic conjecture to that of Cais et al.—one that matches the available data.

本文给出了一种计算有限域Fq上具有给定a数的g属超椭圆曲线所占比例的新方法。在特征三中，该方法给出了形式为Y2=f(X)且f(X)∈Fq[X]的单调和无立方概率曲线的精确概率，与Cais等人在先前工作中提供的数据相匹配。这些结果足以在限制为无平方f时对这些概率（用q表示）进行精确估计。因此，对于正整数a和g，我们证明了由a数为a的曲线组成的g属超椭圆曲线模空间的非空层的余维为2a−1。这与层的协维为a(a+1)/2的阿贝尔变化的模空间的类似结果形成对比。最后，我们的结果允许Cais等人的另一种启发式猜想-一种与可用数据相匹配的猜想。

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引用次数: 0

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