Finite Fields and Their Applications最新文献

In this paper we develop further studies on nilpotent linearized polynomials (NLP's) over finite fields, a class of polynomials introduced by the second author. We characterize certain NLP's that are binomials and show that, in general, NLP's are also nilpotent over a particular tower of finite fields. We also develop results on the construction of permutation polynomials from NLP's, extending some past results. In particular, the latter yields polynomials that permutes certain infinite subfields of ${\bar{F}}_q$ and have a very particular cycle structure. Finally, we provide a nice correspondence between certain NLP's and involutions in binary fields and, in particular, we discuss a general method to produce affine involutions over binary fields without fixed points.

We study four sums including the Jacquet–Piatetski-Shapiro–Shalika, Flicker, Bump–Friedberg, and Jacquet–Shalika sums associated to irreducible cuspidal representations of general linear groups over finite fields. By computing explicitly, we relate Asai and Bump–Friedberg gamma factors over finite fields to those over nonarchimedean local fields through level zero supercuspidal representation. Via Deligne–Kazhdan close field theory, we prove that exterior square and Bump–Friedberg gamma factors agree with corresponding Artin gamma factors of their associated tamely ramified representations through local Langlands correspondence. We also deduce product formul$\ae$ for Asai, Bump–Friedberg, and exterior square gamma factors in terms of Gauss sums. By combining these results, we examine Jacquet–Piatetski-Shapiro–Shalika, Flicker–Rallis, Jacquet–Shalika, and Friedberg–Jacquet periods and vectors and their connections to Rankin–Selberg, Asai, exterior square, and Bump–Friedberg gamma factors, respectively.

Linear complementary dual codes (LCD codes for short) are an important subclass of linear codes which have nice applications in communication systems, cryptography, consumer electronics and information protection. In the literature, it has been proved that an $[n,k,d]$ Euclidean LCD code over $F_q$ with $q>3$ exists if there is an $[n,k,d]$ linear code over $F_q$, where q is a prime power. However, the existence of binary and ternary Euclidean LCD codes has not been totally characterized. Hence it is interesting to construct binary and ternary Euclidean LCD codes with new parameters. In this paper, we construct new families of binary and ternary leading-systematic Euclidean LCD codes from some special functions including semibent functions, quadratic functions, almost bent functions, and planar functions. These LCD codes are not constructed directly from such functions, but come from some self-orthogonal codes constructed with such functions. Compared with known binary and ternary LCD codes, the LCD codes in this paper have new parameters.

A normal element in a finite field extension $F_{q^n}/F_q$ is characterized by having linearly independent conjugates over $F_q$. We consider the generalization of normal elements known as k-normal elements, where a subset of the conjugates are required to be linearly independent. In this paper, we provide an explicit combinatorial formula for counting the number of k-normal elements in a finite field extension motivated by an open problem proposed by Huczynska, Mullen, Panario, and Thomson in 2013. Furthermore, we use these results to establish new insights about $F_q$-practical numbers.

Ahlswede, Khachatrian, Mauduit and Sárközy introduced the notion of family complexity, Gyarmati, Mauduit and Sárközy introduced the cross-correlation measure for families of binary sequences. It is a challenging problem to find families of binary sequences with both small cross-correlation measure and large family complexity. In this paper we present a family of binary sequences with both small cross-correlation measure and large family complexity by using the properties of character sums and primitive normal elements in finite fields.

Let S be a rational fraction and let f be a polynomial over a finite field. Consider the transform $T\left(f\right)=\mathrm{numerator}\left(f\right(S\left)\right)$. In certain cases, the polynomials f, $T\left(f\right)$, $T\left(T\right(f\left)\right)\dots$ are all irreducible. For instance, in odd characteristic, this is the case for the rational fraction $S=(x^2+1)/\left(2x\right)$, known as the R-transform, and for a positive density of irreducible polynomials f. We interpret these transforms in terms of isogenies of elliptic curves. Using complex multiplication theory, we devise algorithms to generate a large number of rational fractions S, each of which yields infinite families of irreducible polynomials for a positive density of starting irreducible polynomials f.

Let $K$ be an algebraically closed field of characteristic $p>0$. A pressing problem in the theory of algebraic curves is the determination of the p-rank of a (nonsingular, projective, irreducible) curve $X$ over $K$. This birational invariant affects arithmetic and geometric properties of $X$, and its fundamental role in the study of the automorphism group $\mathrm{Aut}\left(X\right)$ has been noted by many authors in the past few decades. In this paper, we provide an extensive study of the p-rank of curves of Fermat type $y^m=x^n+1$ over $K={\overline{F}}_p$. We determine a combinatorial formula for this invariant in the general case and show how this leads to explicit formulas of the p-rank of several such curves. By way of illustration, we present explicit formulas for more than twenty subfamilies of such curves, where m and n are generally given in terms of p. We also show how the approach can be used to compute the p-rank of other types of curves.

In this work, we propose two criteria for linear codes obtained from the Plotkin sum construction being symplectic self-orthogonal (SO) and linear complementary dual (LCD). As specific constructions, several classes of symplectic SO codes with good parameters including symplectic maximum distance separable codes are derived via ℓ-intersection pairs of linear codes and generalized Reed-Muller codes. Also symplectic LCD codes are constructed from general linear codes. Furthermore, we obtain some binary symplectic LCD codes, which are equivalent to quaternary trace Hermitian additive complementary dual codes that outperform the best-known quaternary Hermitian LCD codes reported in the literature. In addition, we prove that symplectic SO and LCD codes obtained in these ways are asymptotically good.

This article explores additive codes with one-rank hull, offering key insights and constructions. The article introduces a novel approach to finding one-rank hull codes over finite fields by establishing a connection between self-orthogonal elements and solutions of quadratic forms. It also provides a precise count of self-orthogonal elements for any duality over the finite field $F_q$, particularly odd primes. Additionally, construction methods for small rank hull codes are introduced. The highest possible minimum distance among additive one-rank hull codes is denoted by $d_1{[n,k]}_{p^e,M}$. The value of $d_1{[n,k]}_{p^e,M}$ for $k=1,2$ and $n\ge 2$ with respect to any duality M over any finite field $F_{p^e}$ is determined. Furthermore, the new quaternary one-rank hull codes are identified over non-symmetric dualities with better parameters than symmetric ones.

Suppose that F is a finite field and $R=M_n\left(F\right)$ is the ring of n-square matrices over F. Here we characterize when the Cayley graph of the additive group of R with respect to the set of invertible elements of R, called the unitary Cayley graph of R, is well-covered. Then we apply this to characterize all finite rings with identity whose unitary Cayley graph is well-covered or Cohen-Macaulay.