Euler characteristic of a minimum code trellis is maximal

A class of separable block codes is defined that includes group and linear codes. A code trellis is called minimal if it has the minimal vertex number |V| (the order of code symbols is fixed). It is shown that a minimal trellis of a separable code has the minimal ribs number |E| and the maximum Euler characteristic |V|-|E|. Therefore, the Viterbi decoding complexity of a separable code is minimal in using the minimal code trellis because the Viterbi algorithm demands the performance of the |E| summations and |E|-|V|+1 comparisons.