Finite Fields and Their Applications最新文献

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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 : 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

On small densities defined without pseudorandomness 在没有伪随机性定义的小密度上

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-10-08 DOI: 10.1016/j.ffa.2025.102735

Thomas Karam

We identify a new sufficient condition on linear forms

ϕ_{1}, \dots, ϕ_{k} : F_{p}^{n} \to F_{p}

which guarantees that every subset of

{0, 1}^{n}

on which none of

ϕ_{1}, \dots, ϕ_{k}

has full image has a density which tends to 0 with k. The condition is much weaker than the condition usually used to guarantee that

(ϕ_{1} (x), \dots, ϕ_{k} (x))

takes each value of

F_{p}^{k}

with probability close to

p^{- k}

when x is chosen uniformly at random in the Boolean cube

{0, 1}^{n}

. The density is at most quasipolynomially small in k, a bound that is necessarily close to sharp.

我们在线性形式中确定了一个新的充分条件，它保证在每个{0,1}n的子集上，如果不存在一个完整的图像，则其密度随k趋近于0。该条件远弱于通常用来保证当在布尔立方体{0,1}n中均匀随机选择x时，(ϕ1（x)，…，ϕk(x)）取Fpk的每个值的概率接近p - k的条件。密度在k中最多是准多项式的小，这个边界必然接近于锐。

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 : 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

Automorphism groups of the fields of definition of torsion points of elliptic curves in characteristic ≥5 特征≥5的椭圆曲线扭转点定义域的自同构群

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-10-07 DOI: 10.1016/j.ffa.2025.102737

Bo-Hae Im , Hansol Kim

For a field K of characteristic

p \geq 5

, let

E_{s, t} : y^{2} = x^{3} + s x + t

be an elliptic curve defined over the function field

K (s, t)

in two variables s and t. For a non-negative positive integer e and a positive integer N which is not divisible by p, we prove that if

K \supseteq F_{p}^{alg}

, then the automorphism group of the normal extension

K (s, t) (E_{s, t} [p^{e} N])

over

K (s, t)

is isomorphic to

{(Z / p^{e} Z)}^{\times} \times {SL}_{2} (Z / N Z)

. Applying this result, we also determine the automorphism group of the normal extension

K (s, t) (E_{s, t} [p^{e} N])

for a general field K of characteristic

p \geq 5

对于特征p≥5的域K，设Es，t:y2=x3+sx+t是定义在函数域K（s,t）上的两个变量s和t上的椭圆曲线。对于一个非负正整数e和一个不能被p整除的正整数N，我们证明了如果K(s,t)（Es,t[peN]）在K（s,t）上的正规扩展K（s,t）的自同构群与（Z/peZ）××SL2（Z/NZ）同构。应用这一结果，我们还确定了特征p≥5的一般域K的正规扩展K(s,t)（Es,t[peN]）的自同构群。

引用次数: 0

Brocard-Ramanujan problem for polynomials over finite fields 有限域上多项式的Brocard-Ramanujan问题

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-09-30 DOI: 10.1016/j.ffa.2025.102731

Wataru Takeda

The Brocard-Ramanujan problem is an unsolved number theory problem to find integer solutions

(x, n)

x^{2} - 1 = n!

. In this paper, we consider this problem over polynomial rings

F_{q} [T]

, where

F_{q}

is a finite field with q elements. We find all solutions to the equation

X^{2} - 1 = Π_{C} (n)

, where

Π_{C} (n)

denotes the Carlitz factorial. More precisely, we characterize all solutions and prove that there are infinitely many solutions if and only if

F_{q}

is an extension of

F_{4}

. This characterization is achieved without using the Mason-Stothers theorem, analogous to the abc conjecture for integers.

Brocard-Ramanujan问题是求解x2−1=n!的整数解（x,n）的未解数论问题。本文考虑多项式环Fq[T]上的这一问题，其中Fq是一个有q个元素的有限域。我们找到方程X2−1=ΠC(n)的所有解，其中ΠC(n)表示Carlitz阶乘。更准确地说，我们刻画了所有解，并证明了当且仅当Fq是F4的扩展时存在无穷多个解。这个特征是不使用梅森-斯托瑟斯定理，类似于整数的abc猜想。

引用次数: 0

Trace duality and additive complementary pairs of additive cyclic codes over finite chain rings 有限链环上加性循环码的迹对偶和加性互补对

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-09-29 DOI: 10.1016/j.ffa.2025.102732

Sanjit Bhowmick , Kuntal Deka , Alexandre Fotue Tabue , Edgar Martínez-Moro

This paper investigates the algebraic structure of complementary pairs of additive cyclic codes over a finite commutative chain ring of odd characteristic. We demonstrate that for every additive complementary pair of additive codes, both constituent codes are free modules. Moreover, we present a necessary and sufficient condition for a pair of additive codes over a finite commutative chain ring of odd characteristic to form an additive complementary pair. Finally, we show that, in the case of a complementary pair of additive cyclic codes over a finite chain ring of odd characteristic, one of the codes is permutation equivalent to the trace dual of the other.

研究了奇特征有限交换链环上加性循环码的互补对的代数结构。证明了对于每一对加性码的加性互补对，其组成码都是自由模。此外，我们还给出了奇数特征的有限交换链环上一对加性码形成加性互补对的充分必要条件。最后，我们证明了在奇数特征的有限链环上的加性循环码的互补对中，其中一个码是另一个码的迹对偶的置换等价。

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 : 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

Weierstrass semigroups and automorphism group of a maximal function field with the third largest possible genus, q≡0(mod3) 具有第三大可能属q≡0（mod3）的极大函数域的Weierstrass半群和自同构群

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-09-22 DOI: 10.1016/j.ffa.2025.102729

Peter Beelen , Maria Montanucci , Lara Vicino

In this article, we explicitly determine the Weierstrass semigroup at any place and the full automorphism group of a known

F_{q^{2}}

-maximal function field

Z_{3}

, which is realised as a Galois subfield of the Hermitian function field and has the third largest genus, for

q \equiv 0 (\mod 3)

. This completes the work contained in [3] and [4], where the cases

q \equiv 2 (\mod 3)

and

q \equiv 1 (\mod 3)

, respectively, were studied. Like for these other two cases, the problem of determining the uniqueness of the function field

Z_{3}

, with respect to the value of its genus, is still open. The knowledge of the Weierstrass semigroups may be instrumental in finding a solution to this problem, as it happened to be the case for the function fields with the largest [11] and second largest genera [1], [7]. Similarly to what observed in [3] and [4], also in the case of

Z_{3}

we find that many different types of Weierstrass semigroups appear, and that the set of Weierstrass places contains also non-

F_{q^{2}}

-rational places. We also determine

Aut (Z_{3})

, which turns out to be exactly the automorphism group inherited from the Hermitian function field, apart from the case

q = 3

在这篇文章中，我们明确地确定了在任何地方的Weierstrass半群和已知的fq2 -极大函数域Z3的完全自同构群，它被实现为hermite函数域的伽罗瓦子域，并且具有第三大属，对于q≡0（mod3）。这就完成了[3]和[4]中所包含的工作，其中分别研究了q≡2（mod3）和q≡1（mod3）的情况。就像其他两种情况一样，确定函数域Z3的唯一性的问题，关于它的属的值，仍然是开放的。Weierstrass半群的知识可能有助于找到这个问题的解决方案，因为它恰好是具有最大[11]和第二大属[1]，[7]的函数域的情况。与[3]和[4]的情况类似，在Z3的情况下，我们发现出现了许多不同类型的Weierstrass半群，并且Weierstrass位的集合也包含非fq2 -有理位。我们还确定了Aut(Z3)，除了q=3的情况外，它就是继承自厄米函数场的自同构群。

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引用次数: 0

The dual codes of two families of BCH codes 两个BCH码族的双码

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-09-17 DOI: 10.1016/j.ffa.2025.102721

Haojie Xu , Xia Wu , Wei Lu , Xiwang Cao

In this paper, we present an infinite family of MDS codes over

F_{2^{s}}

and two infinite families of almost MDS codes over

F_{p^{s}}

for any prime p, by investigating the parameters of the dual codes of two families of BCH codes. Notably, these almost MDS codes include two infinite families of near MDS codes over

F_{3^{s}}

, resolving a conjecture posed by Geng et al. in 2022. Furthermore, we demonstrate that both of these almost MDS codes and their dual codes hold infinite families of 3-designs over

F_{p^{s}}

for any prime p. Additionally, we study the subfield subcodes of these families of MDS and near MDS codes, and provide several binary, ternary, and quaternary codes with best known parameters.

本文通过研究两族BCH码的对偶码的参数，给出了任意素数p上F2s上的无限族MDS码和Fps上的两个无限族几乎MDS码。值得注意的是，这些近MDS码包括F3s上的两个无限近MDS码族，解决了耿等人在2022年提出的一个猜想。此外，我们证明了这些几乎MDS码和它们的对偶码在任意素数p的Fps上都具有无限族的3-设计。此外，我们研究了这些MDS和近MDS码族的子域子码，并提供了几种具有最已知参数的二进制、三进制和四进制码。

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引用次数: 0

Mean value theorems for short rational exponential sums 短有理指数和的中值定理

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-09-09 DOI: 10.1016/j.ffa.2025.102719

Doowon Koh , Igor E. Shparlinski

We obtain finite field analogues of a series of recent results on various mean value theorems for Weyl sums. Instead of the Vinogradov Mean Value Theorem, our results rest on the classical argument of Mordell, combined with several other ideas.

我们得到了一系列最近关于Weyl和的各种中值定理的结果的有限域类似物。我们的结果不是基于维诺格拉多夫中值定理，而是基于莫德尔的经典论点，并结合了其他几个观点。

引用次数: 0

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