Minimal trellis diagrams of lattices

Unlike block codes, the number of distinct paths on the trellis diagrams of lattices (N) depends highly on the selected (trellis) coordinate system. Focusing on N as the measure of complexity, it is shown that the problem of finding a proper trellis of a lattice can be reduced to the problem of finding a proper basis for the lattice. For a lattice Λ with coding gain γ and dimension n, a lower bound of the form [γ^n/2] on N is derived. A trellis of Λ is called minimal if it achieves the lower bound (or more generally, if it minimizes N). For many important lattices like Barnes-Wall lattices BW_n, n=2^m, the Leech lattice Λ₂₄, D_n, D_n^⊥, E _n, E_n^⊥, and A_n, n⩽9, we obtain the basis matrices which result in minimal trellis diagrams. For some other lattices like the Coxeter-Todd lattice K₁₂ , and A_n, A_n^⊥, n>9, trellises with small values of N (probably not minimal) are obtained. The constructed trellises, which are novel in many cases, can be employed to efficiently decode the lattices via the Viterbi algorithm