On the generalised rank weights of quasi-cyclic codes

Generalised rank weights were formulated in analogy to Wei's generalised Hamming weights, but for the rank metric. In this paper we study the generalised rank weights of quasi-cyclic codes, a special class of linear codes usually studied for their properties in error correction over the Hamming metric. By using the algebraic structure of quasi-cyclic codes, a new upper bound on the generalised rank weights of quasi-cyclic codes is formulated, which is tighter than the known Singleton bound. Additionally, it is shown that the first generalised rank weight of self-dual \begin{document}$ 1 $\end{document} -generator quasi-cyclic codes is almost completely determined by the choice of \begin{document}$ {\mathbb F}_{q^{m}} $\end{document} .