Finite Fields and Their Applications最新文献

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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 : 2026-02-01 Epub 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

Estimates on the number of rational solutions of Markoff-Hurwitz equations over finite fields 有限域上Markoff-Hurwitz方程有理数解的估计

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2026-02-01 Epub Date: 2025-10-08 DOI: 10.1016/j.ffa.2025.102733

Miriam Abdón , Daniela Alves de Oliveira , Juliane Capaverde , Mariana Pérez , Melina Privitelli

Let N denote the number of solutions to the generalized Markoff-Hurwitz-type equation

{(a_{1} X_{1}^{m} + \dots + a_{n} X_{n}^{m} + a)}^{k} = b X_{1} \dots X_{n}

over the finite field

F_{q}

, where

m, k

are positive integers, and

a, b, a_{i} \in F_{q}^{⁎}

for

i = 1, \dots, n

, with

k, m \geq 2

and

n \geq 3

. Using techniques from algebraic geometry, we provide an estimate for N and establish conditions under which the equation admits solutions where all

X_{i}

are nonzero.

设N表示有限域Fq上广义markoff - hurwitz型方程（a1X1m+⋯+anXnm+a）k=bX1⋯Xn的解的个数，其中m、k为正整数，且对于i=1、…、N，当k、m≥2、N≥3时，a、b、ai∈Fq。利用代数几何的技巧，我们提供了N的估计，并建立了方程允许所有Xi都是非零的解的条件。

引用次数: 0

Construction of three class of at most four-weight binary linear codes and their applications 三类最多四权二进制线性码的构造及其应用

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2026-02-01 Epub Date: 2025-11-04 DOI: 10.1016/j.ffa.2025.102754

Tonghui Zhang , Pinhui Ke , Zuling Chang

Three classes of binary linear codes with at most four nonzero weights were constructed in this paper, in which two of them are projective three-weight codes. As applications, s-sum sets for any odd

s > 1

were constructed.

构造了3类至多有4个非零权值的二元线性码，其中2类为射影三权值码。作为应用，构造了任意奇数s>；1的s和集。

引用次数: 0

Large cyclic subspace codes over finite fields 有限域上的大循环子空间码

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2026-02-01 Epub Date: 2025-09-08 DOI: 10.1016/j.ffa.2025.102722

He Zhang , Chunming Tang , Xiwang Cao , Gaojun Luo

Cyclic subspace codes play a crucial role in random network coding. Designing such cyclic subspace codes with the largest possible code size and minimum distance remains a classical problem. Roth et al. (2018) [28] first investigated optimal cyclic subspace codes via Sidon spaces and proved that the orbit of a Sidon space is an optimal cyclic subspace code with full-length orbit. This paper introduces a new method, namely the intermediate extension field, to construct Sidon spaces and cyclic subspace codes. The main results show that our new codes over intermediate fields have optimal minimum distance and contain more codewords than known constructions. Therefore, this work improves the lower bound of optimal cyclic subspace codes.

循环子空间码在随机网络编码中起着至关重要的作用。设计尽可能大码长和最小距离的循环子空间码仍然是一个经典问题。Roth et al.(2018)[28]首次通过西顿空间研究了最优循环子空间码，证明了西顿空间的轨道是具有全长轨道的最优循环子空间码。本文介绍了构造西顿空间和循环子空间码的一种新方法，即中间可拓域。主要结果表明，我们的新码在中间域上具有最佳的最小距离，并且比已知结构包含更多的码字。因此，本文改进了最优循环子空间码的下界。

引用次数: 0

On the study of semi-involutory and semi-orthogonal matrices 半对合矩阵和半正交矩阵的研究

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2026-02-01 Epub Date: 2025-09-29 DOI: 10.1016/j.ffa.2025.102730

Yogesh Kumar , Susanta Samanta , P.R. Mishra , Atul Gaur

This paper investigates semi-involutory and semi-orthogonal matrices, presenting two algorithms for verifying these properties for

n \times n

matrices over

F_{p^{m}}^{⁎}

. The algorithms significantly reduce computational complexity by avoiding the need for non-singular diagonal matrices. The structure of circulant matrices with these properties is also explored, including the derivation of the exact form of their corresponding diagonal matrices. We further investigate MDS matrices with semi-involutory and semi-orthogonal properties. We prove that semi-involutory circulant matrices of order

n \geq 3

over

F_{2^{m}}

, where

\gcd (n, 2^{m} - 1) = 1

, cannot be MDS matrices. Additionally, we show that semi-orthogonal circulant matrices of order

2^{n}

over

F_{2^{m}}

cannot be MDS. Finally, explicit formulas for counting

3 \times 3

semi-involutory and semi-orthogonal MDS matrices over

F_{2^{m}}

are derived, and exact counts for

4 \times 4

semi-involutory and semi-orthogonal MDS matrices over

F_{2^{m}}

for

m = 3

and 4 are provided.

本文研究了半对合矩阵和半正交矩阵，给出了验证Fpm上n×n矩阵的这些性质的两种算法。该算法避免了对非奇异对角矩阵的需要，大大降低了计算复杂度。本文还探讨了具有这些性质的循环矩阵的结构，包括其对应对角矩阵的精确形式的推导。我们进一步研究了具有半对合和半正交性质的MDS矩阵。证明了n≥3阶/ F2m的半对合循环矩阵，其中gcd (n,2m−1)=1不能是MDS矩阵。此外，我们还证明了2n / F2m阶的半正交循环矩阵不可能是MDS。最后，导出了F2m上3×3半对合和半正交MDS矩阵的显式计数公式，并给出了m=3和4时F2m上4×4半对合和半正交MDS矩阵的精确计数。

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 : 2026-02-01 Epub 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

The multiplicity-one theorem for the superspeciality of curves of genus two 二属曲线超特性的重性- 1定理

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2026-02-01 Epub Date: 2025-10-10 DOI: 10.1016/j.ffa.2025.102738

Shushi Harashita, Yuya Yamamoto

Igusa proved in 1958 that the polynomial determining the supersingularity of elliptic curves in Legendre form is separable. In this paper, we get an analogous result for curves of genus 2 in Rosenhain form. More precisely we show that the ideal determining the superspeciality of the curve has multiplicity one at every superspecial point. Igusa used a Picard-Fucks differential operator annihilating a Gauß hypergeometric series. We shall use the Lauricella system (of type D) of hypergeometric differential equations in three variables.

Igusa在1958年证明了决定勒让德形式椭圆曲线超奇异性的多项式是可分离的。本文对Rosenhain形式的2属曲线得到了一个类似的结果。更确切地说，我们证明了确定曲线超特性的理想在每一个超特殊点上都具有多重性。Igusa使用了一个Picard-Fucks微分算子来湮灭高斯超几何级数。我们将使用三变量超几何微分方程的Lauricella系统（D型）。

引用次数: 0

Linear l-intersection pairs of matrix-product codes and their applications 矩阵积码的线性l交对及其应用

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2026-02-01 Epub Date: 2025-11-10 DOI: 10.1016/j.ffa.2025.102758

Xiujing Zheng, Sujuan Huang, Shixin Zhu

The linear l-intersection pairs of codes are a generalization of linear complementary dual (LCD) codes, hulls of codes and linear complementary pairs (LCPs) of codes. Matrix-product codes are extended versions derived from shorter codes through matrix-product techniques. In this paper, we investigate linear l-intersection pairs of matrix-product codes. The characterization of these pairs can be achieved by examining the dimension of the intersection between their respective constituent codes. For the dimension part of the conjecture for linear l-intersection pairs of codes proposed by Guenda et al. (Des Codes Cryptogr. 88: 133-152, 2020), we prove that if the conjecture holds for prime lengths, then its dimension part holds. As a practical application, linear l-intersection pairs of matrix-product codes are utilized to the constructions of asymmetric quantum error-correcting (AQEC) codes and asymmetric entanglement-assisted quantum error-correcting (AEAQEC) codes. Some instances exhibit favorable parameters.

线性l交码对是线性互补对偶码、码壳和码的线性互补对的推广。矩阵积代码是通过矩阵积技术从较短的代码衍生而来的扩展版本。本文研究了矩阵积码的线性l交对。这些对的特征可以通过检查它们各自组成代码之间的交集的维度来实现。对于Guenda et al. （Des codes Cryptogr. 88: 133- 152,2020）提出的线性l交码对猜想的维数部分，证明了如果该猜想对于素数长度成立，则其维数部分成立。在实际应用中，将线性l交对矩阵积码用于构造非对称量子纠错码和非对称纠缠辅助量子纠错码。有些实例显示出有利的参数。

引用次数: 0

On S-complete mappings for large S 大S的S完全映射

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2026-02-01 Epub Date: 2025-11-07 DOI: 10.1016/j.ffa.2025.102756

Robert S. Coulter , Paul Hearding

Given a subset S of a finite field, an S-complete mapping is a polynomial

f (X)

for which

f (X) + c X

is a permutation polynomial over the finite field for each

c \in S

. In this paper, we introduce a new method for constructing permutation polynomials and use it to establish a class of S-complete mappings with “large” S.

给定一个有限域的子集S，一个S完全映射是一个多项式f(X)，其中f(X)+cX是有限域上对每个c∈S的一个置换多项式。本文提出了一种构造置换多项式的新方法，并利用它建立了一类具有“大”S的S-完全映射。

引用次数: 0

Permutation polynomials and finite projective spaces 置换多项式与有限射影空间

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2026-02-01 Epub Date: 2025-11-05 DOI: 10.1016/j.ffa.2025.102752

Tong Lin , Qiang Wang

Using arbitrary bases for the finite field

F_{q^{n}}

over

F_{q}

, we obtain the generalized Möbius transformations (GMTs), which are a class of bijections between the projective geometry

PG (n - 1, q)

and the set of roots of unity

μ_{\frac{q^{n} - 1}{q - 1}} \subseteq F_{q^{n}}

, where

n \geq 2

is any integer. We also introduce a class of projective polynomials, using the properties of which we determine the inverses of the GMTs. Moreover, we study the roots of those projective polynomials, which lead to a three-way correspondence between partitions of

F_{q^{n}}^{⁎}, μ_{\frac{q^{n} - 1}{q - 1}}

and

PG (n - 1, q)

. Through this correspondence and the GMTs, we construct permutation polynomials of index

\frac{q^{n} - 1}{q - 1}

over

F_{q^{n}}

利用有限域Fqn / Fq上的任意基，得到了广义Möbius变换（GMTs），它是投影几何PG（n−1,q）与单位μqn−1q−1的根集合之间的一类双射，其中n≥2为任意整数。我们还引入了一类射影多项式，利用它们的性质来确定gmt的逆。此外，我们还研究了这些射影多项式的根，得到了Fqn _，μqn−1q−1和PG（n−1,q）的分区之间的三向对应关系。通过这种对应关系和GMTs，我们构造了指标qn−1q−1 / Fqn的置换多项式。

引用次数: 0

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