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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 positiveinteger and $p$ a prime number, are investigated. By the Chinese RemainderTheorem, structural properties and the decomposition of GQC codes are given.For 1-generator GQC codes, minimal generating sets and lower bounds on theminimum 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 thispaper.

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