The trellis structure of maximal fixed-cost codes

We show that the family of maximal fixed-cost (MFC) codes, with codeword costs defined in a right-cancellative semigroup, are rectangular, and hence admit biproper trellis presentations. Among all possible trellis presentations for a rectangular code, biproper trellises minimize a wide variety of complexity measures, including the Viterbi decoding complexity. Examples of MFC codes include such “nonlinear” codes as permutation codes and shells of constant norm in the integer lattice, as well as linear codes over a finite field. The intersection of two rectangular codes is another rectangular code; therefore, “nonlinear” codes such as lattice shells or words of constant weight in a linear code have biproper trellis presentations. We show that every rectangular code can be interpreted as an MFC code. Applications of these results include error detection, trellis-based indexing, and soft-decision decoding