Application of symbolic dynamics to the study of trellis group codes
N.T. Sindhushayana · 1996
A number of good, non-linear trellis codes that are used in practice are known to out-perform any linear code with the same structural complexity. Typically, these codes possess a great deal of symmetry in them. This work emerged from an attempt to characterize such codes, that not linear per se, but nevertheless exhibit many of the symmetry properties of linear codes. Our work draws from Slepian's theory of finite group codes and Forney's notion of geometrically uniform signal sets. The distinctive feature of our approach involves the characterization of trellis codes from the point of view of Symbolic Dynamics. Our main thesis is that geometrically uniform trellis codes for the Gaussian channel are infinite Slepian-type codes, whose codewords consist of sequences over the n-dimensional Euclidean space $\IR\sp{n}$. Such a code is described by a symbolic system over a finite group, which acts on a finite set of labels arising from an isometric labeling of a (possibly infinite) signal set in $\IR\sp{n}$. This theory results in a natural description of geometrically uniform, rotationally invariant trellis codes as the class of codes that arise from symbolic systems which contain a certain constant sequence. In this thesis, we present the general theory of symbolic dynamics, along with some new results which play a central role in the study of trellis group codes. We develop a classification scheme for general symbolic systems, which provides a conceptual framework for the characterization of trellis group codes. We introduce the notion of group systems and orbit systems, which establish the bridge between symbolic dynamics and trellis coded modulation. We present an algorithm to obtain the minimal graph of a group system from a finite list of cycles that generate them. We also present a new, inductive technique to determine the symmetries of a large class of trellis codes. Finally, we present certain results on rotationally invariant trellis group codes based on PSK signal sets.