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.
Categories: cs.IT math.IT