On the trellis structure of block codes

Due to the editorial process, there may be slight discrepancies between this version and the published one. Abstract|The problem of minimizing the vertex count at a given time index in the trellis for a general (nonlinear) code is shown to be NP-complete. Examples are provided that show that 1) the minimal trellis for a nonlinear code may not be observable, i.e., some codewords may be represented by more than one path through the trellis and 2) minimizing the vertex count at one time index may be incompatible with minimizing the vertex count at another time index. A trellis product is deened and used to construct trellises for sum codes. Minimal trellises for linear codes are obtained by forming the product of elementary trellises corresponding to the one-dimensional subcodes generated by atomic codewords. The structure of the resulting trellis is determined solely by the spans of the atomic codewords. A correspondence between minimal linear block code trellises and conngurations of non-attacking rooks on a triangular chess board is established and used to show that the number of distinct minimal linear block code trellises is a Stirling number of the second kind. Various bounds on trellis size are re-interpreted in this context .meration.