Finite Fields and Their Applications最新文献

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The automorphism group of the pn-torsion points of an elliptic curve over a field of characteristic p ≥ 5

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-04-22 DOI: 10.1016/j.ffa.2025.102631

Bo-Hae Im , Hansol Kim

For a field K of characteristic

p \geq 5

and the elliptic curve

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

defined over the function field

K (s, t)

of two variables s and t, we prove that for a positive integer n, the automorphism group of the normal extension

K (s, t) (E_{s, t} [p^{n}]) / K (s, t)

is isomorphic to

{(Z / p^{n} Z)}^{\times}

, and its inseparable degree is

p^{n}

引用次数: 0

(σ,τ)-derivations of group rings with applications

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-04-22 DOI: 10.1016/j.ffa.2025.102629

Praveen Manju , Rajendra Kumar Sharma

<div><div>Leo Creedon and Kieran Hughes in [18] studied derivations of a group ring RG (of a group G over a commutative unital ring R) in terms of generators and relators of group G. In this article, we do that for <math><mo>(</mo><mi>σ</mi><mo>,</mo><mi>τ</mi><mo>)</mo></math>-derivations. We develop a necessary and sufficient condition such that a map <math><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>R</mi><mi>G</mi></math> can be extended uniquely to a <math><mo>(</mo><mi>σ</mi><mo>,</mo><mi>τ</mi><mo>)</mo></math>-derivation D of RG, where R is a commutative ring with unity, G is a group having a presentation <math><mo>〈</mo><mi>X</mi><mo>|</mo><mi>Y</mi><mo>〉</mo></math> (X the set of generators and Y the set of relators) and <math><mo>(</mo><mi>σ</mi><mo>,</mo><mi>τ</mi><mo>)</mo></math> is a pair of R-algebra endomorphisms of RG which are R-linear extensions of the group endomorphisms of G. Further, we classify all inner <math><mo>(</mo><mi>σ</mi><mo>,</mo><mi>τ</mi><mo>)</mo></math>-derivations of the group algebra RG of an arbitrary group G over an arbitrary commutative unital ring R in terms of the rank and a basis of the corresponding R-module consisting of all inner <math><mo>(</mo><mi>σ</mi><mo>,</mo><mi>τ</mi><mo>)</mo></math>-derivations of RG. We obtain several corollaries, particularly when G is a <math><mo>(</mo><mi>σ</mi><mo>,</mo><mi>τ</mi><mo>)</mo></math>-FC group or a finite group G and when R is a field. We also prove that if R is a unital ring and G is a group whose order is invertible in R, then every <math><mo>(</mo><mi>σ</mi><mo>,</mo><mi>τ</mi><mo>)</mo></math>-derivation of RG is inner. We apply the results obtained above to study σ-derivations of commutative group algebras over a field of positive characteristic and to classify all inner and outer σ-derivations of dihedral group algebras <math><mi>F</mi><msub><mrow><mi>D</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msub></math> (<math><msub><mrow><mi>D</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msub><mo>=</mo><mo>〈</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>|</mo><msup><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>=</mo><msup><mrow><mi>b</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mn>1</mn><mo>,</mo><msup><mrow><mi>b</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mi>a</mi><mi>b</mi><mo>=</mo><msup><mrow><mi>a</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>〉</mo></math>, <math><mi>n</mi><mo>≥</mo><mn>3</mn></math>) over an arbitrary field <math><mi>F</mi></math> of any characteristic. Finally, we g

引用次数: 0

The minimum distance of a parameterized code over an even cycle 参数化代码在偶数周期上的最小距离

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-04-15 DOI: 10.1016/j.ffa.2025.102630

Eduardo Camps-Moreno , Jorge Neves , Eliseo Sarmiento

Parameterized linear codes over a graph exhibit a very interesting relation between their basic parameters and the combinatorics of the graph. We address the computation of the most elusive of all parameters, the minimum distance. We focus on the case of the parameterized code of order 1 over an even cycle.

图上的参数化线性编码在其基本参数和图的组合之间表现出一种非常有趣的关系。我们将讨论所有参数中最难以捉摸的参数--最小距离--的计算。我们重点研究偶数循环上阶数为 1 的参数化代码。

引用次数: 0

Stephen D. Cohen's contributions to the finite fields community

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-04-10 DOI: 10.1016/j.ffa.2025.102627

Sophie Huczynska, Gary L. Mullen

In this short note, to mark the retirement of Stephen D. Cohen from his editorial role at this journal, we discuss and acknowledge his significant contributions to the finite fields community in the areas of service and research.

引用次数: 0

Cyclically covering subspaces of Fqn

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-04-10 DOI: 10.1016/j.ffa.2025.102625

Meng Sun, Changli Ma, Liwei Zeng

Let q be a prime power, n be a positive integer, and

F_{q}^{n}

be the n-dimensional row vector space over finite field

F_{q}

. We say a subspace U of

F_{q}^{n}

is cyclically covering if the union of the cyclic shifts of U is equal to

F_{q}^{n}

. Recently, the largest possible codimension of a cyclically covering subspace of

F_{q}^{n}

, denoted by

h_{q} (n)

, has attracted the attention of many scholars. In this paper, we introduce cyclically covering subspaces of finite field

F_{q^{n}}

. By virtue of the theory of direct sum decomposition of finite fields, we describe a method for constructing cyclically covering subspaces of

F_{q^{n}}

, and determine the value of

h_{2} (n)

for some special n. In particular, we prove

h_{2} (21) = 4

. Finally, several lower bounds of

h_{q} (n)

are given when

(q, n) = 1

, which generalizes results of the existing results in [2].

引用次数: 0

The study on three classes of permutation quadrinomials over finite fields of odd characteristic

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-04-10 DOI: 10.1016/j.ffa.2025.102626

Changhui Chen , Haibin Kan , Jie Peng , Hengtai Wang , Lijing Zheng

Let p be an odd prime and n a positive integer. This paper focuses on the investigation of permutation quadrinomials with generalized Niho exponents over finite fields of odd characteristic. Inspired by the works in [10], we study two classes of permutation quadrinomials over

F_{p^{n}}

and a class of permutation quadrinomials over

F_{5^{n}}

. Finally, we verify that permutation quadrinomials presented in this paper are multiplicative inequivalent to known ones.

引用次数: 0

Some classes of permutation pentanomials

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-03-27 DOI: 10.1016/j.ffa.2025.102619

Zhiguo Ding , Michael E. Zieve

For each prime

p \neq 3

and each power

q = p^{k}

, we present two large classes of permutation polynomials over

F_{q^{2}}

of the form

X^{r} B (X^{q - 1})

which have at most five terms, where

B (X)

is a polynomial with coefficients in

{1, - 1}

. The special case

p = 2

of our results comprises a vast generalization of 76 recent results and conjectures in the literature. In case

p > 2

, no instances of our permutation polynomials have appeared in the literature, and the construction of such polynomials had been posed as an open problem. Our proofs are short and involve no computations, in contrast to the proofs of many of the special cases of our results which were published previously.

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引用次数: 0

On self-dual completely regular codes with covering radius ρ ≤ 3 关于覆盖半径 ρ ≤ 3 的自双完全正则码

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-03-27 DOI: 10.1016/j.ffa.2025.102617

J. Borges , V. Zinoviev

We give a complete classification of self-dual completely regular codes with covering radius

ρ \leq 3

. For

ρ = 1

the results are almost trivial. For

ρ = 2

, by using properties of the more general class of uniformly packed codes in the wide sense, we show that there are two sporadic such codes, of length 8, and an infinite family, of length 4, apart from the direct sum of two self-dual completely regular codes with

ρ = 1

, each one. For

ρ = 3

, in some cases, we use similar techniques to the ones used for

ρ = 2

. However, for some other cases we use different methods, namely, the Pless power moments which allow to us to discard several possibilities. We show that there are only two self-dual completely regular codes with

ρ = 3

and

d \geq 3

, which are both ternary: the extended ternary Golay code and the direct sum of three ternary Hamming codes of length 4. Therefore, any self-dual completely regular code with

d \geq 3

and

ρ = 3

is ternary and has length 12.

We provide the intersection arrays for all such codes.

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引用次数: 0

The differential uniformity of the power functions xpn+52 over Fpn

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-03-27 DOI: 10.1016/j.ffa.2025.102622

Wenping Yuan , Xiaoni Du , Huan Zhou , Xingbin Qiao

Cryptographic functions with low differential uniformity have important applications in designing S-box in the block ciphers. In this paper, we mainly investigate the differential uniformity

Δ_{F}

on a new class of power mappings

F (x) = x^{\frac{p^{n} + 5}{2}}

over

F_{p^{n}}

with p being an odd prime and n being a positive integer. More precisely, for

p = 3

, the differential uniformity and the differential spectrum of F have been determined explicitly. The results indicate that F is a locally-PN function with differentially

\frac{3^{n} + 1}{4}

-uniform when n is odd and a locally-APN function with differentially

\frac{3^{n} + 3}{4}

-uniform when n is even. Then, for

p = 5

, we prove that F is APN for even n and

Δ_{F} = 6

for odd n through specific differential equations and quadratic character over

F_{5^{n}}

. The method is different from the existing one. Moreover, for primes

p > 5

, we show that

Δ_{F} \leq 5

when

p^{n} \equiv 3 (\mod 4)

and

Δ_{F} \leq 8

when

p^{n} \equiv 1 (\mod 4)

引用次数: 0

Zeros of a system of diagonal polynomials over finite fields

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-03-26 DOI: 10.1016/j.ffa.2025.102623

Yulu Feng

<div><div>Let <math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math> be the finite field of characteristic p, having <math><mi>q</mi><mo>=</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>a</mi></mrow></msup></math> elements and let <math><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup></math> be the unit group of <math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math>. Let <math><mi>N</mi><mo>(</mo><mi>V</mi><mo>)</mo></math> be the number of <math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math>-rational points of the affine algebraic variety defined by the simultaneous vanishing of the diagonal polynomials <math><msub><mrow><mi>f</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mn>1</mn></mrow></msub><msubsup><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi><mn>1</mn></mrow></msub></mrow></msubsup><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mi>n</mi></mrow></msub><msubsup><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi><mi>n</mi></mrow></msub></mrow></msubsup><mo>+</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>i</mi></mrow></msub></math>, <math><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>r</mi></math>, where <math><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>∈</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup></math>, <math><msub><mrow><mi>c</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math> and <math><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></math> is a nonnegative integer for <math><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>r</mi><mo>,</mo><mn>1</mn><mo>≤</mo><mi>j</mi><mo>≤</mo><mi>n</mi></math>. By using properties of Teichmüller representations and the Stickelberger relation applied by Ax and Wan, we show that<math><msub><mrow><mi>ord</mi></mrow><mrow><mi>q</mi></mrow></msub><mi>N</mi><mo>(</mo><mi>V</mi><mo>)</mo><mo>≥</mo><mo>⌈</mo><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mfrac><mrow><mn>1</mn></mrow><mrow><munder><mi>max</mi><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>r</mi></mrow></munder><mo>⁡</mo><mo>{</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>}</mo></mrow></mfrac><mo>⌉</mo><mo>−</mo><mi>r</mi></math> if <math><msub><mrow><mi>max</mi></mrow><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>r</mi></mrow></msub><mo>⁡</mo><mo>{</mo><msub><mrow><mi>d</mi></mrow><

引用次数: 0

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