Permutation trinomials over Fq3

We consider four classes of polynomials over the fields ${\mathbb{F}}_{q^3}$, $q=p^h$, $p>3$, $f_1(x)=x^{q^2+q-1}+A{x}^{q^2-q+1}+Bx$, $f_2(x)=x^{q^2+q-1}+A{x}^{q^3-q^2+q}+Bx$, $f_3(x)=x^{q^2+q-1}+A{x}^{q^2}-Bx$, $f_4(x)=x^{q^2+q-1}+A{x}^q-Bx$, where $A,B\in {\mathbb{F}}_q$. We find sufficient conditions on the pairs $(A,B)$ for which these polynomials permute ${\mathbb{F}}_{q^3}$ and we give lower bounds on the number of such pairs.     

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 $f(x)=x^3+a{x}^{q+2}+b{x}^{2q+1}+c{x}^{3q}$, where $a,b,c\in {\mathbb{F}}_q^{⁎}$, is a permutation quadrinomial of ${\mathbb{F}}_{q^2}$ 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 $char({\mathbb{F}}_q)=2$ 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 $x^3+a{x}^{q+2}+b{x}^{2q+1}+c{x}^{3q}$, where $char({\mathbb{F}}_q)=3,5$ and $a,b,c\in {\mathbb{F}}_q^{⁎}$ and proposed some new classes of permutation quadrinomials of ${\mathbb{F}}_{q^2}$.In particular, in this paper we classify all permutation polynomials of ${\mathbb{F}}_{q^2}$ of the form $f(x)=x^3+a{x}^{q+2}+b{x}^{2q+1}+c{x}^{3q}$, where $a,b,c\in {\mathbb{F}}_q^{⁎}$, over all finite fields of odd characteristic and obtain several new classes of such permutation quadrinomials.     

Perfect squares representing the number of rational points on elliptic curves over finite field extensions

Let q be a perfect power of a prime number p and $E({\mathbb{F}}_q)$ be an elliptic curve over ${\mathbb{F}}_q$ given by the equation $y^2=x^3+Ax+B$. For a positive integer n we denote by $\mathrm{\#}E({\mathbb{F}}_{q^n})$ the number of rational points on E (including infinity) over the extension ${\mathbb{F}}_{q^n}$. Under a mild technical condition, we show that the sequence ${\{\mathrm{\#}E({\mathbb{F}}_{q^n})\}}_{n>0}$ contains at most 10²⁰⁰ perfect squares. If the mild condition is not satisfied, then $\mathrm{\#}E({\mathbb{F}}_{q^n})$ is a perfect square for infinitely many n including all the multiples of 12. Our proof uses a quantitative version of the Subspace Theorem. We also find all the perfect squares for all such sequences in the range $q<50$ and $n\le 1000$.     

Construction and enumeration of left dihedral codes satisfying certain duality properties

Let ${\mathbb{F}}_q$ be the finite field of q elements and let $D_{2n}=〈x,y|x^n=1,y^2=1,yxy=x^{n?1}〉$ be the dihedral group of 2n elements. Left ideals of the group algebra ${\mathbb{F}}_q[D_{2n}]$ are known as left dihedral codes over ${\mathbb{F}}_q$ of length 2n, and abbreviated as left $D_{2n}$-codes. Let $\gcd (n,q)=1$. In this paper, we give an explicit representation for the Euclidean hull of every left $D_{2n}$-code over ${\mathbb{F}}_q$. On this basis, we determine all distinct Euclidean LCD codes and Euclidean self-orthogonal codes which are left $D_{2n}$-codes over ${\mathbb{F}}_q$. In particular, we provide an explicit representation and a precise enumeration for these two subclasses of left $D_{2n}$-codes and self-dual left $D_{2n}$-codes, respectively. Moreover, we give a direct and simple method for determining the encoder (generator matrix) of any left $D_{2n}$-code over ${\mathbb{F}}_q$, and present several numerical examples to illustrative our applications.     

Intersections between the norm-trace curve and some low degree curves

In this paper we analyze the intersection between the norm-trace curve over ${\mathbb{F}}_{q^3}$ and the curves of the form $y=a{x}^3+b{x}^2+cx+d$, giving a complete characterization of the intersection between the curve and the parabolas (a=0), as well as sharp bounds for the other cases. This information is used for the determination of the weight distribution of some one-point AG codes arising from the curve.     

Asymptotically good ZprZps-additive cyclic codes

We construct a class of ${\mathbb{Z}}_{p^r}{\mathbb{Z}}_{p^s}$-additive cyclic codes generated by pairs of polynomials, where p is a prime number. Based on probabilistic arguments, we determine the asymptotic rates and relative distances of this class of codes: the asymptotic Gilbert-Varshamov bound at $\frac{1+p^{s-r}}{2}\delta$ is greater than $\frac{1}{2}$ and the relative distance of the code is convergent to δ, while the rate is convergent to $\frac{1}{1+p^{s-r}}$ for $0<\delta <\frac{1}{1+p^{s-r}}$ and $1\le r<s$. As a consequence, we prove that there exist numerous asymptotically good ${\mathbb{Z}}_{p^r}{\mathbb{Z}}_{p^s}$-additive cyclic codes.     

New results on permutation binomials of finite fields

After a brief review of the existing results on permutation binomials of finite fields, we introduce the notion of equivalence among permutation binomials (PBs) and describe how to bring a PB to its canonical form under equivalence. We then focus on PBs of ${\mathbb{F}}_{q^2}$ of the form $X^n(X^{d(q-1)}+a)$, where n and d are positive integers and $a\in {\mathbb{F}}_{q^2}^{⁎}$. Our contributions include two nonexistence results: (1) If q is even and sufficiently large and $a^{q+1}\ne 1$, then $X^n(X^{3(q-1)}+a)$ is not a PB of ${\mathbb{F}}_{q^2}$. (2) If $2\le d|q+1$, q is sufficiently large and $a^{q+1}\ne 1$, then $X^n(X^{d(q-1)}+a)$ is not a PB of ${\mathbb{F}}_{q^2}$ under certain additional conditions. (1) partially confirms a recent conjecture by Tu et al. (2) is an extension of a previous result with $n=1$.