Fields Of Odd Characteristic Research Articles

Binary Sequences With a Low Correlation via Cyclotomic Function Fields of Odd Characteristic

Binary Sequences With a Low Correlation via Cyclotomic Function Fields of Odd Characteristic

The Complete Differential Spectrum of a Class of Power Permutations Over Odd Characteristic Finite Fields

The Complete Differential Spectrum of a Class of Power Permutations Over Odd Characteristic Finite Fields

Minimal linear codes derived from weakly regular bent and plateaued functions

The construction of (minimal) linear codes from functions has received much attention in the literature. In this paper, we derive several minimal codes from the sets of pre-images of weakly regular plateaued and bent functions over the odd characteristic finite fields. Based on the recent construction method, we obtain six-weight and seven-weight minimal codes from these functions. The Hamming weights in proposed codes are calculated from the character sums of the sets based on the Walsh spectrum of the employed functions, while the weight distributions of the codes are determined from both the sizes of the employed sets and the Walsh distributions of the employed functions.

Classification of some quadrinomials over finite fields of odd characteristic

In this paper, we completely determine all necessary and sufficient conditions such that the polynomial f(x)=x3+axq+2+bx2q+1+cx3q, where a,b,c∈Fq⁎, is a permutation quadrinomial of Fq2 over any finite field of odd characteristic. This quadrinomial has been studied first in [25] by Tu, Zeng and Helleseth, later in [24] Tu, Liu and Zeng revisited these quadrinomials and they proposed a more comprehensive characterization of the coefficients that results with new permutation quadrinomials, where char(Fq)=2 and finally, in [16], Li, Qu, Li and Chen proved that the sufficient condition given in [24] is also necessary and thus completed the solution in even characteristic case. In [6] Gupta studied the permutation properties of the polynomial x3+axq+2+bx2q+1+cx3q, where char(Fq)=3,5 and a,b,c∈Fq⁎ and proposed some new classes of permutation quadrinomials of Fq2.In particular, in this paper we classify all permutation polynomials of Fq2 of the form f(x)=x3+axq+2+bx2q+1+cx3q, where a,b,c∈Fq⁎, over all finite fields of odd characteristic and obtain several new classes of such permutation quadrinomials.

Periodic Groups Saturated with Finite Simple Symplectic Groups of Dimension 6 over Fields of Odd Characteristics

Periodic Groups Saturated with Finite Simple Symplectic Groups of Dimension 6 over Fields of Odd Characteristics

A class of three-weight linear codes over finite fields of odd characteristic

A class of three-weight linear codes over finite fields of odd characteristic

On the pronormality of subgroups of odd index in some direct products of finite groups

A subgroup [Formula: see text] of a group [Formula: see text] is said to be pronormal in [Formula: see text] if [Formula: see text] and [Formula: see text] are conjugate in [Formula: see text] for each [Formula: see text]. Some problems in Finite Group Theory, Combinatorics and Permutation Group Theory were solved in terms of pronormality, therefore, the question of pronormality of a given subgroup in a given group is of interest. Subgroups of odd index in finite groups satisfy a native necessary condition of pronormality. In this paper, we continue investigations on pronormality of subgroups of odd index and consider the pronormality question for subgroups of odd index in some direct products of finite groups. In particular, in this paper, we prove that the subgroups of odd index are pronormal in the direct product [Formula: see text] of finite simple symplectic groups over fields of odd characteristics if and only if the subgroups of odd index are pronormal in each direct factor of [Formula: see text]. Moreover, deciding the pronormality of a given subgroup of odd index in the direct product of simple symplectic groups over fields of odd characteristics is reducible to deciding the pronormality of some subgroup [Formula: see text] of odd index in a subgroup of [Formula: see text], where each [Formula: see text] acts naturally on [Formula: see text], such that [Formula: see text] projects onto [Formula: see text]. Thus, in this paper, we obtain a criterion of pronormality of a subgroup [Formula: see text] of odd index in a subgroup of [Formula: see text], where each [Formula: see text] is a prime and each [Formula: see text] acts naturally on [Formula: see text], such that [Formula: see text] projects onto [Formula: see text].

Construction of minimal linear codes with few weights from weakly regular plateaued functions

The construction of (minimal) linear codes from functions over finite fields has been greatly studied in the literature, since determining the parameters of codes based on functions is rather easy due to the nice structures of functions. In this paper, we derive $3$-weight and $4$-weight linear codes from weakly regular plateaued functions in the recent construction method of linear codes over the odd characteristic finite fields. The Hamming weights and weight distributions for codes are, respectively, determined by using the Walsh transform values and Walsh distributions of the employed functions. We also derive projective $3$-weight punctured codes with good parameters from the constructed codes. These punctured codes may be almost optimal due to the Griesmer bound, and they can be employed to design association schemes. We lastly show that all constructed codes are minimal, which approves that they can be employed to design high democratic secret sharing schemes.

On a class of bent, near-bent, and 2-plateaued functions over finite fields of odd characteristic

<abstract><p>The main purpose of this paper is to study a class of the $ p $-ary functions $ f_{\lambda, u, v}(x) = Tr_1^k(\lambda x^{p^k+1})+Tr^n_1(ux)Tr_1^n(vx) $ for any odd prime $ p $ and $ n = 2k, \lambda\in GF(p^k)^*, u, v\in GF(p^n)^*. $ With the help of Fourier transforms, we are able to subdivide the class of all $ f_{\lambda, u, v} $ into sublcasses of bent, near-bent and 2-plateaued functions. It is shown that the choice of $ \lambda, u $ and $ v $, ensuring that $ f $ is bent, 2-plateaued or near-bent, is directly related to finding the subset $ A\subset GF(p)^3 $. The efficient method for defining the set $ A\subset GF(p)^3 $ is described in detail.</p></abstract>

Secondary constructions of (non)weakly regular plateaued functions over finite fields

Plateaued (vectorial) functions over finite fields have diverse applications in symmetric cryptography, coding theory, and sequence theory. Constructing these functions is an attractive research topic in the literature. We can distinguish two kinds of constructions of plateaued functions: secondary constructions and primary constructions. The first method uses already known functions to obtain new functions while the latter do not need to use previously constructed functions to obtain new functions. In this work, the first secondary constructions of (non)weakly regular plateaued (vectorial) functions are presented over the finite fields of odd characteristics. We also introduce some recursive constructions of (non)weakly regular plateaued $p$-ary functions by using already known such functions. We obtain nontrivial plateaued functions from the previously known trivial plateaued (partially bent) functions in the proposed construction methods.

Algorithms for quadratic forms over global function fields of odd characteristic

This paper presents an adaptation of recently developed algorithms for quadratic forms over number fields in [4] to global function fields of odd characteristics. First, we present algorithm for checking if a given non-degenerate quadratic form is isotropic or hyperbolic. Next we devise a method for computing the dimension of the anisotropic part of a quadratic form. Finally we present algorithms computing two field invariants: the level and the Pythagoras number.

Cyclic Codes from a Sequence over Finite Fields

A cyclic code has been one of the most active research topics in coding theory because they have many applications in data storage systems and communication systems. They have efficient encoding and decoding algorithms. This paper explains the construction of a family of cyclic codes from sequences generated by a trace of a monomial over finite fields of odd characteristics. The parameter and some examples of the codes are presented in this paper.

Characterization of nilpotent Lie algebras of breadth 3 over finite fields of odd characteristic

In this paper, we give a characterization of finite-dimensional nilpotent Lie algebras of breadth 3 over finite fields of odd characteristic. This characterization parallels to the one for finite p-groups of breadth 3 given earlier in [5].

More results about a class of quadrinomials over finite fields of odd characteristic

Let denote the finite field with q elements. In this paper, we study the permutation property of the class of quadrinomials where With some additional assumption on the coefficients, we propose several new necessary and sufficient conditions on the coefficients that yield permutation quadrinomials.

Construction of Binary Sequences With Low Correlation via Multiplicative Quadratic Character Over Finite Fields of Odd Characteristics

In literature, there are several methods to construct Gold sequences. One of the constructions is via the trace function from extension field of \mathbb F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> . Estimation of correlation of this construction is based on number of rational points of elliptic curves. In this article, we generalize this construction from finite fields of even characteristic to odd characteristics by using multiplicative quadratic character. Again, estimation of correlation of this construction is based on number of rational points of elliptic curves. Thus, we obtain binary sequences which have more flexibility on length while still possessing low correlation property. Moreover, some of the sequences are optimally balanced.

Notes about the linear complexity of cyclotomic binary sequences of order four over a finite field of odd characteristic

In this paper, we investigate the linear complexity and the minimal polynomial of cyclotomic binary sequences of order four over a finite field of odd characteristic. The sequences considered are determined on the basis of two cyclotomic classes of the fourth order. We show that they have a high linear complexity.

A surface in odd characteristic with discrete and non-finitely generated automorphism group

It was proved by Tien-Cuong Dinh and me that there is a smooth complex projective surface whose automorphism group is discrete and not finitely generated. In this paper, after observing finite generation of the automorphism group of any smooth projective surface birational to any K3 surface over any algebraic closure of the prime field of odd characteristic, we will show that there is a smooth projective surface, birational to some K3 surface, such that the automorphism group is discrete and not finitely generated, over any algebraically closed field of odd characteristic of positive transcendental degree over the prime field.

The large sieve inequality with square moduli for quadratic extensions of function fields

In this paper, we establish a version of the large sieve inequality with square moduli for imaginary quadratic extensions of rational function fields of odd characteristics.

A note on the distinctness of some Kloosterman sums

The Fischer result about distinctness of the Kloosterman sums on F∗p is extended for the finite fields of degrees of extension that are powers of 2. To obtain the desired outcome, we give an elementary proof of the fact that there does not exist a pair of Kloosterman sums on same odd characteristic fields which are opposite to each other.

A recurrent construction of irreducible polynomials of fixed degree over finite fields

In this paper we consider in detail the composition of an irreducible polynomial with X^2 and suggest a recurrent construction of irreducible polynomials of fixed degree over finite fields of odd characteristics. More precisely, given an irreducible polynomial of degree n and order 2^rt with t odd, the construction produces ord_t(2) irreducible polynomials of degree n and order t. The construction can be used for example to search irreducible polynomials with specific requirements on its coefficients.