Finite Fields and Their Applications最新文献

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Binary polycyclic codes associated with x2η+1+x2η+1: Hamming distance, duality, reversibility and LCD properties 与x2η+1+x2η+1相关的二进制多循环码：汉明距离、对偶性、可逆性和LCD性质

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-10-17 DOI: 10.1016/j.ffa.2025.102741

Sujata Bansal, Pramod Kumar Kewat

This work explores binary polycyclic codes associated with the polynomial

x^{2^{η + 1}} + x^{2^{η}} + 1

, which is the

2^{η}

-th power of

x^{2} + x + 1

for every integer

η \geq 1

. We provide an in-depth structural analysis of these codes and compute the exact Hamming distance of each of these binary polycyclic codes. Furthermore, we determine the parity-check matrices and examine the Euclidean duals and annihilator duals for these polycyclic codes. Our analysis reveals that these codes are reversible and, in certain cases, are Linear Complementary Dual (LCD) codes. This discovery highlights the potential of these codes in practical applications such as communication systems, data storage, consumer electronics, and cryptography. We also propose a conjecture that suggests all such polycyclic codes can be LCD.

本文研究了与多项式x2η+1+x2η+1相关的二进制多环码，对于每个η≥1的整数，它是x2+x+1的2η-次幂。我们对这些码进行了深入的结构分析，并计算了每个二进制多环码的精确汉明距离。进一步，我们确定了这些多环码的奇偶校验矩阵，并检验了它们的欧几里得对偶和湮灭对偶。我们的分析表明，这些代码是可逆的，在某些情况下，是线性互补双（LCD）代码。这一发现突出了这些代码在通信系统、数据存储、消费电子和密码学等实际应用中的潜力。我们还提出了一个猜想，表明所有这些多循环码都可以是LCD。

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引用次数: 0

Irreducible factorizations of polynomials xpk+1−bx+b over a finite field 有限域上多项式xpk+1 - bx+b的不可约分解

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-10-16 DOI: 10.1016/j.ffa.2025.102740

Xue Jia , Fengwei Li , Huan Sun , Qin Yue

In this paper, we investigate polynomials of the form

f (x) = x^{p^{k} + 1} - b x + b

, where

0 \neq b \in F_{p^{n}}

, p is a prime, and k divides n. By introducing a new approach based on the projective general linear group, we show that the number of zeros of

f (x)

F_{p^{n}}

belongs to

{0, 1, 2, p^{k} + 1}

, and provide explicit criteria on b for each case. We also count the number of such polynomials corresponding to each possible number of zeros. Moreover, for the cases where

f (x)

has at least one zero, we determine its complete irreducible factorization over

F_{p^{n}}

本文研究了形式为f(x)=xpk+1 - bx+b的多项式，其中0≠b∈Fpn， p是素数，k除n。通过引入一种基于射影一般线性群的新方法，证明了f(x)在Fpn中的零个数属于{0,1,2,pk+1}，并给出了每种情况下b的显式判据。我们还计算对应于每个可能的零数的多项式的个数。此外，对于f(x)至少有一个零的情况，我们确定了它在Fpn上的完全不可约分解。

引用次数: 0

An unusual family of supersingular curves of genus five in characteristic two 特征二的五属超奇异曲线的一个不寻常的族

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-10-10 DOI: 10.1016/j.ffa.2025.102736

Dušan Dragutinović

We construct a family of smooth supersingular curves of genus 5 in characteristic 2 with several notable features: its dimension matches the expected dimension of any component of the supersingular locus in genus 5, its members are non-hyperelliptic curves with non-trivial automorphism groups, and each curve in the family admits a double cover structure over both an elliptic curve and a genus-2 curve. We also provide an explicit parametrization of this family.

构造了特征2上的5属光滑超奇异曲线族，它的维数与5属超奇异轨迹的任意分量的期望维数相匹配，它的成员是具有非平凡自同构群的非超椭圆曲线，族中的每条曲线都在椭圆曲线和2属曲线上具有双覆盖结构。我们还提供了这个家族的显式参数化。

引用次数: 0

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 : 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

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

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