Multidimensional Quasi-Cyclic and Convolutional Codes

Abstract

We introduce multidimensional generalizations of quasi-cyclic codes and investigate their algebraic properties as well as their links to multidimensional convolutional codes. We call these generalized codes n-dimensional quasi-cyclic (QnDC) codes. We provide a concatenated structure for QnDC codes in the sense that they can be decomposed into shorter codes over extensions of their base field. This structure allows us to prove that these codes are asymptotically good. Then, we extend the relation between quasi-cyclic and convolutional codes to multidimensional case. Lally has shown that the free distance of a convolutional code is lower bounded by the minimum distance of an associated quasi-cyclic code. We show that a QnDC code can be associated to a given nD convolutional code. Moreover, we prove that the relation between distances of convolutional and quasicyclic codes extend to a class of 1-generator 2D convolutional codes and the associated Q2DC codes. Along the way, an alternative new description of noncatastrophic polynomial encoders is given for 1-generator 1D convolutional codes and a sufficient condition for noncatastrophic nD polynomial encoders is obtained for 1-generator nD convolutional codes.

A survey on product codes and 2-D codes

“…Encoding and decoding problems for multidimensional cyclic codes along with their structure have been discussed extensively by Saints in [31]. Güneri and Özkaya studied the structure of multidimensional analogues of quasi-cyclic codes in [28]. Also, generalizations of various problems discussed in this survey have already been provided.…”

The Concatenated Structure of Quasi-Abelian Codes

“…where C m i denotes the cyclic group Z/m i Z of order m i for i = 1, 2, then an H-QA code can be viewed as a QC code of co-index m 1 or co-index m 2 . Moreover, as mentioned before in [11] and [15] for certain special cases, we have various QA structures with different indices for a given QA code, since an…”

“…Jitman and Ling reconsidered QA codes ( [17]) giving, among other things, the decomposition of QA codes by extending the CRT decomposition for QC introduced by Ling-Solé (see also [6]). Let us note that a special class of QA codes is also studied in [11], which is called by the authors multidimensional QC codes, or quasi nD cyclic codes. In the general case, the family of QA codes coincides with the family QC codes, but if one considers strictly QA codes, these form a proper subfamily of QC codes.…”