Canonical representation of quasi-cyclic codes

A linear block code C of length n is called quasi-cyclic (QC) if it is invariant under a cyclic shift of L positions, T^L, where L<n. Any cyclic code can be represented by a unique generator polynomial. In this paper we associate with QC-codes a polynomial generator set which is a natural generalization of the generator polynomial of a cyclic code. A canonical generator matrix of a QC-code which is invariant under T^L is introduced which shows the symmetric structure of the n/L-section minimal trellis diagram (MTD). The state space dimension is nondecreasing on the left half of this trellis. The canonical generator matrix is important because it provides considerable information about the trellis complexity of QC codes as well as the relation between these codes and convolutional codes