Applied and Computational Mathematics Seminar

Title: Convolutional Codes Over Semisimple Word-Ambients

Speaker: Steve Szabo, Department of Mathematics, Ohio University

Abstract: A fundamental division in Coding Theory classifies codes as being Block Codes or Convolutional Codes. All Block Codes may be viewed as being Convolutional in a trivial way but the converse is not true. Among the most studied families of block codes are the cyclic codes, which may be thought of as being the ideals of a specific semisimple quotient ring of the ring of polynomials with coefficients in the alphabet field of the code. An approach to extending the notion of cyclicity to Convolutional Codes leads to viewing cyclic convolutional as certain left ideal direct summands of an automorphism-skewed polynomial ring. In the most general sense, a skew polynomial ring includes not only an automorphism but also a derivation on the ring of coefficients. Up to this point, the literature on cyclic convolutional codes has not considered setting them in this more general context. Other line of enquiry where only very limited work has been done is considering other identities (outside from cyclicity) whose study may be extended from the block code arena to the convolutional setting. Many of these additional identities place the block codes of interest as ideals of certain semisimple rings (i.e. negacyclic, constacyclic, group codes, etc.) In this talk, the notion of convolutional codes over semisimple word-ambients will be investigated where the skew polynomial ring includes a derivation. Our specific goal is to establish conditions on the semisimple ambient and on the proposed automorphism and derivations to guarantee that non-trivial (i.e. non-block) convolutional codes exist.