Dissertation Defense

Abstract: The notion of cyclic convolutional codes is extended to describe a larger family of codes. This family also includes the group convolutional codes. The ingredients to create such codes are a semisimple artinian algebra A (the word ambient) as well as an automorphism sigma on A and a sigma-derivation delta. Conditions on sigma and delta for the existence of non-block convolutional codes are given. In general, the convolutional codes we study here are certain left ideals of the general skew polynomial ring R=A[z;sigma;delta]. It is shown that for commutative word ambients the induced convolutional codes are principal left ideals of R. The techniques from the theory of cyclic convolutional codes are expanded to this new setting and used to provide a matrix based view of group convolutional codes. This approach allows us to produce duals for certain group convolutional codes.

For certain alphabet finite rings R with characteristic p, repeated root codes of length p^s have been studied under some additional hypotheses. However, up until now, some gaps remained in the literature. In this work, all remaining cases of cyclic and negacyclic codes of length p^s over a Galois ring alphabet are considered. A method for computing the Hamming distance of these codes is provided. In addition, the general structure of the code ambient for polycyclic codes over Galois rings is studied.