On Locally Invertible Encoders and Multidimensional Convolutional Codes

Abstract

Multidimensional (m-D) convolutional codes generalize the well known notion of a 1-D convolutional code defined over a univariate polynomial ring with coefficients in a finite field to multivariate polynomial rings. The more complicated structure of a multivariate polynomial ring when compared to a univariate one, however, makes the generalization nontrivial. While 1-D convolutional codes have been thoroughly understood and have wide applications in communication systems, the theory of m-D convolutional codes is still in its infancy, and these codes lack unified notation and practical implementation.

This dissertation develops a sequence space approach for realizing m-D convolutional codes. While most of the existing research is focused on algebraic aspects, fundamental issues regarding practical implementation that are well developed and fairly straightforward in the 1-D case have remained undefined for m-D convolutional codes. In this dissertation we address some of these issues.

We define a new notion of sequence space ordering and show that certain multivariate polynomial matrices which we call as locally invertible encoders, when transformed to the sequence space domain, have an invertible subsequence map between their input and output sequences. This subsequence map has a well defined structure that allows for the explicit construction of locally invertible encoders by performing elementary operations on the ground field without the use of any polynomial operations. We use the invertible subsequence map to introduce a novel method to encode and invert multidimensional sequences. We show that locally invertible encoders have good structural properties which make them a natural choice to generate multidimensional convolutional codes.