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 Rⁿ as the orbit of a bi-infinite “seed sequence”, x₀∈(R ⁿ)^Z, 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 8-state, rotationally invariant trellis code designed by Wei (1984), used in the V.32 (and V.32 bis) international modem standard, belongs to this class