Trellis group codes for the Gaussian channel

In this paper, trellis group codes are introduced as an extension of Slepian group codes to codes over sequence spaces. A trellis group code is defined over R” as the orbit of a bi-infinite “seed sequence”, 20 E (W”)‘, under an infinite, defining group of transformations. This group of transformations is generated by a symbolic system. The theory is developed by combining a nontrivial extension of the notion of an isometric labeling, with results from the theory of symbolic dynamics over groups. New results presented here include a useful characterization of uniform partitions and a symbolic dynamic classification of trellis group codes. The theory is used to develop a class of rotationally invariant, nonabelian trellis group codes for QAM modulation. It is also shown that the S-state, rotationally invariant trellis code designed by Wei, used in the V.32 (and V.32 bis) international modem standard, belongs to this class. these symmetries are sufficiently rich as to generate C. This duality of viewpoints leads one to consider two problems: the “synthesis problem,” or how does one generate “good” group codes, and the “analysis problem,” or, how does one decide if a given code is a group code? Forney [2] opened these two problems to the important and much broader class of “trellis codes.” For purposes of this paper, a trellis code is an infinite-dimensional code, described in terms of bi-infinite sequences, based on component block codes lying in W”. Forney’s observations suggest many open questions, several of which are addressed in this paper.