Rank q-cyclic and pseudo-q-cyclic codes

In the theory of codes based on Hamming metric, three classes of codes are well known: cyclic codes, shortened cyclic codes and pseudo-cyclic codes. An important result is that the class of linear shortened cyclic codes coincides with the class of linear pseudo-cyclic codes. No similar results are known in the theory of rank-metric based codes. In this paper, we generalize the notion of q-cyclic codes and introduce two new families of codes, namely, shortened q-cyclic codes and pseudo-q-cyclic codes. It is proved that the class of pseudo-q-cyclic codes coincides with the class of shortened q-cyclic codes if the number of positions to shorten is a multiple of the extension degree. The problem is still open for other shortening.