Algebraic Structure Theory of Tail-Biting Trellises
It is well known that there is an intimate connection between algebraic descriptions of linear block codes in the form of generator or parity-check matrices, and combinatorial descriptions in the form of trellises. A conventional trellis for a linear code C is a directed labelled layered graph with unique start and final nodes, and all paths from the start to the final node spell out codewords. The trellis can be thought of as being laid out on a linear time axis. There is a rich theory of conventional trellises for linear block codes. Every linear block code has a unique minimal trellis, and several seemingly different constructions proposed, all yield this minimal trellis, which simultaneously minimizes all measures of trellis complexity. Tail-biting trellises are defined on circular time axes, and the underlying theory is a little more involved as there is no unique minimal trellis. Interestingly, the complexity of a tail-biting trellis can be much lower than that of the best possible conventional trellis. We extend the well-known BCJR construction for conventional trellises to linear tail-biting trellises, introducing the notion of a displacement matrix. This implicitly induces a coset decomposition of the code. The BCJR-like labeling scheme yields a very simple specification for the tail-biting trellis for the dual code, with the dual trellis having the same statecomplexity profile as that of the primal code . We also show that the algebraic specification of Forney for state spaces of conventional trellises has a natural extension to tail-biting trellises. Finally we provide an automata-theoretic view of trellises and display some connections between well known results in finite automata and trellis theory.