On quintic quasi-cyclic codes

Quasi-cyclic codes of length 5ℓ5ℓ and index ℓℓ over FqFq are obtained from a pair of codes over, respectively, FqFq and Fq4Fq4, by a combinatorial construction called here the quintic construction. They enjoy a designed trellis description and a suboptimal coset decoding algorithm. They are shown to be cyclic when the component codes are cyclic of odd length coprime to 55. Extremal binary self-dual quintic codes are constructed in lengths 60 and 70 by a randomized algorithm.