Generalized Quasi-Cyclic Codes Over ${\mathbb{F}}_q+u{\mathbb{F}}_q$

Generalized quasi-cyclic (GQC) codes with arbitrary lengths over the ring ${\mathbb{F}}_q+u{\mathbb{F}}_q$, where $u^2=0$, $q=p^n$, $n$ a positive integer and $p$ a prime number, are investigated. By the Chinese Remainder Theorem, structural properties and the decomposition of GQC codes are given. For 1-generator GQC codes, minimal generating sets and lower bounds on the minimum distance are given. As a special class of GQC codes, quasi-cyclic (QC) codes over ${\mathbb{F}}_q+u{\mathbb{F}}_q$ are also discussed briefly in this paper.