The Skew generalized quasi-cyclic codes over non-chain ring $F_{q}+\mathfrak{v}F_{q}$}

SGQC codes over a non-chain ring

Abstract

For a prime $p$, let $F_q$ be the finite field of order $q=p^d$. This paper presents the study on skew generalized quasi-cyclic ($\texttt{SGQC}$) codes of length $n$ over the non-chain ring $F_q+\mathfrak{v}F_q$, where $\mathfrak{v}^2=\mathfrak{v}$. Here, first, we prove the dual of an $\texttt{SGQC}$ code of length $n$ is also an $\texttt{SGQC}$ code of the same length and derive a necessary and sufficient condition for the existence of a self-dual code. Then, we discuss the $1$-generator polynomial and the $\rho$-generator polynomial for these codes. Further, we determine the dimension and BCH-type bound for the $1$-generator case. As a by-product, with the help of MAGMA software, we provide a few examples of $\texttt{SGQC}$ codes with improved parameters compared to those given in the existing literature and obtain some $2$-generator optimal and near-optimal $\texttt{SGQC}$ codes of index $2$.