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.     

On the equational graphs over finite fields

In this paper, we generalize the notion of functional graph. Specifically, given an equation $E(X,Y)=0$ with variables X and Y over a finite field ${\mathbb{F}}_q$ of odd characteristic, we define a digraph by choosing the elements in ${\mathbb{F}}_q$ as vertices and drawing an edge from x to y if and only if $E(x,y)=0$. We call this graph as equational graph. In this paper, we study the equational graph when choosing $E(X,Y)=(Y^2-f(X))(\lambda Y^2-f(X))$ with $f(X)$ a polynomial over ${\mathbb{F}}_q$ and λ a non-square element in ${\mathbb{F}}_q$. We show that if f is a permutation polynomial over ${\mathbb{F}}_q$, then every connected component of the graph has a Hamiltonian cycle. Moreover, these Hamiltonian cycles can be used to construct balancing binary sequences. By making computations for permutation polynomials f of low degree, it appears that almost all these graphs are strongly connected, and there are many Hamiltonian cycles in such a graph if it is connected.     

Non-integrated defect relation for meromorphic mappings on Kähler manifolds intersecting non-subdegenerate families of hyperplanes

Let M be a complete Kähler manifold, whose universal covering is biholomorphic to a ball ${\mathbb{B}}^m(R_0)$ in ${\mathbb{C}}^m$ ($0<R_0\le +\infty$). Our purpose of this article is to establish a non-integrated defect relation with truncated level for a meromorphic mapping on M intersecting a family of hyperplanes in ${\mathbb{P}}^n(\mathbb{C})$ which is non-subdegenerate with respect to the mapping.     

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 $n=1$.