Multidimensional Quasi-Cyclic and Convolutional Codes

Abstract

We introduce multidimensional analogues of quasi-cyclic (QC) codes and study their algebraic structure. We demonstrate a concatenated structure for multidimensional QC codes and use this to prove that this class of codes is asymptotically good. We also relate the new family of codes to convolutional codes. It is known that the minimum distance of QC codes provides a natural lower bound on the free distance of convolutional codes. We show that the same relation also holds between certain rank one 2-D convolutional codes and the related multidimensional QC codes. We provide examples, which show that our bound is sharp in some cases. We also present some optimal 2-D QC codes. Along the way, we provide a condition on the encoders of rank one convolutional codes, which are equivalent to noncatastrophicity for 1-D convolutional codes. In the $n\text{D}$ case ( $n>1$ ), our condition is sufficient for the noncatastrophicity of the encoder.