The Factorization Theorem and new algebraic insights into the theory of linear trellises

Abstract

We present a new algebraic framework for linear trellises which yields a new and simpler proof of the fundamental Factorization Theorem by Koetter and Vardy [4], and which sheds light on several other foundational questions that were not considered before. The techniques used within this framework are new, and comprise algebraic tools that can be used to analyze systematically the structure of linear trellises. In fact our methods and tools produce several subsequent results, the most important of which are: characterization of linear trellises isomorphy, uniqueness of linear structure of linearizable trellises, methods for determining all possible factorizations of trellises. These same algebraic methods have a potential for extending the Factorization Theorem to the case of group trellises. In fact this is very important, since the Factorization Theorem for group trellises as formulated by Koetter and Vardy is false.