The Euler Characteristic of the Minimal Code Trellis is Maximum

A class of separable block codes is defined. The class includes group and linear codes. A code trellis is called the minimal trellis if it has the minimum number of vertices $|V|$| (the order of code symbols is fixed). We show that the minimal trellis of a separable code minimizes the edge count $|E|$ and maximizes the Euler characteristic $|V|-|E|$. Thus, the Viterbi decoding complexity of a separable code is minimum when it is implemented on the minimal code trellis, since the Viterbi decoding algorithm requires $|E|$ additions and $|E|-|V|+1$ comparisons.