Extremal Type II Z4-codes constructed from binary doubly even self-dual codes of length 40

In this note, we demonstrate that every binary doubly even self-dual code of length $40$ can be realized as the residue code of some extremal Type II ${\mathbb{Z}}_4$-code. As a consequence, it is shown that there are at least $94356$ inequivalent extremal Type II ${\mathbb{Z}}_4$-codes of length $40$.     

A class of minimal cyclic codes over finite fields

Let ${\mathbb{F}}_q$ be a finite field of odd order $q$ and $n=2^a{p}_1^{a_1}{p}_2^{a_2}$, where $a,a_1,a_2$ are positive integers, $p_1,p_2$ are distinct odd primes and $4p_1{p}_2|q?1$. In this paper, we study the irreducible factorization of $x^n?1$ over ${\mathbb{F}}_q$ and all primitive idempotents in the ring ${\mathbb{F}}_q\left[x\right]∕〈x^n?1〉$.Moreover, we obtain the dimensions and the minimum Hamming distances of all irreducible cyclic codes of length $n$ over ${\mathbb{F}}_q$.     

On the additive cyclic structure of quasi-cyclic codes

An index $?$, length $m?$ quasi-cyclic code can be viewed as a cyclic code of length $m$ over the field ${\mathbb{F}}_{q^{?}}$ via a basis of the extension ${\mathbb{F}}_{q^{?}}∕{\mathbb{F}}_q$. However, this cyclic code is only linear over ${\mathbb{F}}_q$, making it an additive cyclic code, or an ${\mathbb{F}}_q$-linear cyclic code, over the alphabet ${\mathbb{F}}_{q^{?}}$. This approach was recently used in Shi et al. (2017) [16] to study a class of quasi-cyclic codes, and more importantly in Shi et al. (2017) [17] to settle a long-standing question on the asymptotic performance of cyclic codes. Here, we answer one of the problems posed in these two articles, and characterize those quasi-cyclic codes which have ${\mathbb{F}}_{q^{?}}$-linear cyclic images under a basis of the extension ${\mathbb{F}}_{q^{?}}∕{\mathbb{F}}_q$. Our characterizations are based on the module structure of quasi-cyclic codes, as well as on their CRT decompositions into constituents. In the case of a polynomial basis, we characterize the constituents by using the theory of invariant subspaces of operators. We also observe that analogous results extend to the case of quasi-twisted codes.     

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.     

Uniqueness of Fq-quadratic perfect nonlinear maps from Fq3 to Fq2

Let q be a power of an odd prime. We prove that all ${\mathbb{F}}_q$-quadratic perfect nonlinear maps from ${\mathbb{F}}_{q^3}$ to ${\mathbb{F}}_q^2$ are equivalent. We also give a geometric method to find the corresponding equivalence explicitly.     

Graphical characterization of positive definite non symmetric quasi-Cartan matrices

It is known that each positive definite quasi-Cartan matrix $A$ is $\mathbb{Z}$-equivalent to a Cartan matrix $A_{\Delta }$ called Dynkin type of $A$, the matrix $A_{\Delta }$ is uniquely determined up to conjugation by permutation matrices. However, in most of the cases, it is not possible to determine the Dynkin type of a given connected quasi-Cartan matrix by simple inspection. In this paper, we give a graph theoretical characterization of non-symmetric connected quasi-Cartan matrices. For this purpose, a special assemblage of blocks is introduced. This result complements the approach proposed by Barot (1999, 2001), for ${\mathbb{A}}_n$, ${\mathbb{D}}_n$ and ${\mathbb{E}}_m$ with $m=6,7,8$.