Finite Fields and Their Applications最新文献

英文中文

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 : 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

Weierstrass semigroups and automorphism group of a maximal function field with the third largest possible genus, q ≡ 1 (mod 3) 具有第三大可能属q的极大函数域的Weierstrass半群和自同构群 ≡ 1 （mod 3）

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-07-21 DOI: 10.1016/j.ffa.2025.102701

Peter Beelen , Maria Montanucci , Lara Vicino

In this article we continue the work started in [3], explicitly determining the Weierstrass semigroup at any place and the full automorphism group of a known

F_{q^{2}}

-maximal function field

Y_{3}

having the third largest genus, for

q \equiv 1 (\mod 3)

. This function field arises as a Galois subfield of the Hermitian function field, and its uniqueness (with respect to the value of its genus) is a well-known open problem. Knowing the Weierstrass semigroups may provide a key towards solving this problem. Surprisingly enough,

Y_{3}

has many different types of Weierstrass semigroups and the set of its Weierstrass places is much richer than its set of

F_{q^{2}}

-rational places. We show that a similar exceptional behaviour does not occur in terms of automorphisms, that is,

Aut (Y_{3})

is exactly the automorphism group inherited from the Hermitian function field, apart from small values of q.

在这篇文章中，我们继续[3]中开始的工作，明确地确定在任何地方的Weierstrass半群和已知的具有第三大属的fq2 -极大函数域Y3的完全自同构群，对于q≡1（mod3）。这个函数场是厄米函数场的伽罗瓦子场，它的唯一性（相对于它的格值）是一个众所周知的开放问题。了解weerstrass半群可能是解决这个问题的关键。令人惊讶的是，Y3有许多不同类型的Weierstrass半群而且它的Weierstrass位的集合比它的fq2 -有理位的集合要丰富得多。我们证明了在自同构方面不会出现类似的异常行为，即，除了q的小值外，Aut（Y3）正是从厄米函数场继承的自同构群。

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引用次数: 0

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 : 2025-07-17 DOI: 10.1016/j.ffa.2025.102700

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

引用次数: 0

On iso-dual MDS codes from elliptic curves 椭圆曲线上的等对偶MDS码

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-07-16 DOI: 10.1016/j.ffa.2025.102699

Yunlong Zhu , Chang-An Zhao

For a linear code C over a finite field, if its dual code

C^{⊥}

is equivalent to itself, then the code C is said to be isometry-dual. In this paper, we first confirm a conjecture about the isometry-dual MDS elliptic codes proposed by Han and Ren. Subsequently, two constructions of isometry-dual maximum distance separable (MDS) codes from elliptic curves are presented. The new code length n satisfies

n \leq \frac{q + ⌊ 2 \sqrt{q} ⌋ - 1}{2}

when q is even and

n \leq \frac{q + ⌊ 2 \sqrt{q} ⌋ - 3}{2}

when q is odd. Additionally, we consider the hull dimension of both constructions. In the case of finite fields with even characteristics, an isometry-dual MDS code is equivalent to a self-dual MDS code and a linear complementary dual MDS code. Finally, we apply our results to entanglement-assisted quantum error correcting codes (EAQECCs) and obtain two new families of MDS EAQECCs.

对于有限域上的线性码C，如果它的对偶码C⊥等于它本身，那么这个码C就被称为等距对偶。本文首先证实了Han和Ren提出的关于等距对偶MDS椭圆码的一个猜想。随后，给出了椭圆曲线上的两种等距-对偶最大距离可分离码的构造。当q为偶数时，新码长n满足n≤q+⌊2q⌋−12；当q为奇数时，新码长n满足n≤q+⌊2q⌋−32。此外，我们还考虑了两种结构的船体尺寸。在偶特征有限域的情况下，等距对偶MDS码等价于自对偶MDS码和线性互补对偶MDS码。最后，我们将我们的结果应用于纠缠辅助量子纠错码（EAQECCs），并获得了两个新的MDS EAQECCs家族。

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引用次数: 0

Evaluations of Eisenstein sums 爱森斯坦和的求值

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-07-11 DOI: 10.1016/j.ffa.2025.102698

Ron Evans , Mark Van Veen

Let

F_{q}

be a field of

q = p^{r}

elements. For

k = 9, 11, 13, 14, 15, 16, 18, 20, 22, 23, 26, 30, 46

we evaluate Eisenstein sums of order k over

F_{q}

when r is the smallest positive integer for which

p^{r} \equiv 1 (\mod k)

设Fq是一个由q=pr个元素组成的域。Fork=9,11,13,14,15,16,18,20,22,23,26,30,46当r是pr≡1（modk）的最小正整数时，我们求k / Fq阶的爱森斯坦和。

引用次数: 0

Closed formulas for the generators of all constacyclic codes and for the factorization of Xn − 1, the n-th cyclotomic polynomial and every composition of the form f(Xn) over a finite field for arbitrary positive integers n 所有常环码的生成和Xn的分解的封闭公式 − 1，第n个环多项式和任意正整数n在有限域上的形式f（Xn）的每一个组合

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-07-11 DOI: 10.1016/j.ffa.2025.102695

Anna-Maurin Graner

In this paper we present the solution to four 150 year-old closely related problems in the study of polynomials and algebraic codes over a finite field

F_{q}

. We give one closed formula each for the factorization of the polynomials

X^{n} - a

for arbitrary

a \in F_{q}^{⁎}

X^{n} - 1

, the n-th cyclotomic polynomial and the composition

f (X^{n})

for arbitrary monic irreducible polynomials

f \in F_{q} [X]

f \neq X

, for arbitrary positive integers n. With a new perspective on these problems we show that the factorization of

X^{n} - a

has a beautiful underlying structure which is completely determined by the order of a in the multiplicative group

F_{q}^{⁎}

The binomial

X^{n} - a

and the composition

f (X^{n})

were first considered over prime fields by Joseph Alfred Serret in 1866. In recent years, the factorization of

X^{n} - a

has been studied extensively due to the fact that its factors are the generators of the popular constacyclic codes over finite fields. The factorization of

X^{n} - 1

and of its famous factor, the n-th cyclotomic polynomial, was first studied over prime fields by Carl Friedrich Gauss (among others) in the middle of the 19th century. Since then, many mathematicians were fascinated by this factorization and nowadays it is needed for the construction of the cyclic codes over finite fields.

Many formulas for the factorization of the four polynomials for special choices of n and a were obtained by a large number of mathematicians. Our formulas naturally extend, cover, combine and complete all of these partial solutions.

本文给出了有限域Fq上多项式和代数码研究中四个有150年历史的密切相关问题的解。对于任意正整数n，对于任意a∈Fq， Xn−1，对于任意单不可约多项式f∈Fq[X]， f≠X，我们分别给出了多项式Xn−a的因式分解和复合f（Xn）的封闭公式。我们从一个新的角度证明了Xn−a的因式分解具有一个美丽的内在结构，它完全由乘法群Fq中a的阶数决定。二项式Xn−a和组成f（Xn）是由Joseph Alfred Serret在1866年首次提出的。近年来，由于Xn−a的因式分解是有限域上常见的常环码的生成因子，因此对Xn−a的因式分解得到了广泛的研究。19世纪中期，卡尔·弗里德里希·高斯（Carl Friedrich Gauss）首先在素数场上研究了Xn−1及其著名的因子n-环多项式的分解。从那时起，许多数学家对这种分解着迷，现在需要它来构造有限域上的循环码。对于n和a的特殊选择，许多数学家得到了四个多项式的因式分解的许多公式。我们的公式自然地扩展、涵盖、组合并完成所有这些部分解。

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引用次数: 0

Square values of several polynomials over a finite field 有限域上若干多项式的平方值

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-07-11 DOI: 10.1016/j.ffa.2025.102696

Kaloyan Slavov

Let

f_{1}, \dots, f_{m}

be polynomials in n variables with coefficients in a finite field

F_{q}

. We estimate the number of points

\underline{x}

F_{q}^{n}

such that each value

f_{i} (\underline{x})

is a nonzero square in

F_{q}

. The error term is especially small when the

f_{i}

define smooth projective quadrics with nonsingular intersections. We improve the error term in a recent work by Asgarli–Yip on mutual position of smooth quadrics.

设f1，…，fm是有限域Fq中n个变量的多项式。我们估计Fqn中x_的个数，使得每个值fi（x_）是Fq中的非零平方。当定义具有非奇异交点的光滑投影二次曲面时，误差项特别小。在Asgarli-Yip最近关于光滑二次曲面互位的研究中，我们改进了误差项。

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 : 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

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 : 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

Counting irreducible polynomials with restricted coefficients 计数具有限制系数的不可约多项式

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-07-07 DOI: 10.1016/j.ffa.2025.102691

Kaimin Cheng

Let q be a prime power, and let

F_{q}

denote the finite field with q elements. Consider a positive integer n, and let

R = {R_{i}}_{i = 0}^{n - 1}

be a family of subsets of

F_{q}

. Define

N (R, n)

as the number of monic irreducible polynomials of degree n over

F_{q}

where the coefficient of each non-leading term

T^{i}

lies in

F_{q} ∖ R_{i}

. In this paper, we provide an asymptotic formula for

N (R, n)

, extending a result of Porritt to a more general case.

设q是一个素数幂，设Fq表示有q个元素的有限域。考虑一个正整数n，设R={Ri}i=0n−1是Fq的子集族。定义N（R， N）为N / Fq次的不可约一元多项式的个数，其中每个非前导项Ti的系数在Fq∈Ri中。本文给出了N（R， N）的渐近公式，将Porritt的结果推广到更一般的情况。

引用次数: 0

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