The trellis structure of maximal fixed-cost codes

We show that the family of maximal fixed-cost codes, with codeword costs defined in a right-cancellative semigroup, have biproper trellis presentations. Examples of maximal fixed-cost codes include such "nonlinear" codes as permutation codes, shells of constant Euclidean norm in the integer lattice, and of course ordinary linear codes over a finite field. The intersection of two codes having biproper trellis presentations is another code with a biproper trellis presentation; therefore "nonlinear" codes such as lattice shells or words of constant weight in a linear code have biproper trellis presentations.