A constructive approach for reducing the state complexity of self-dual lattices

This work presents a systematic method to successively minimize the state complexity of the self-dual lattices. This is based on representing the lattice on an orthogonal co-ordinate system corresponding to the Gram-Schmidt (GS) vectors of a Korkin-Zolotarev (KZ) reduced basis. We give expressions for the GS vectors of a KZ basis of the E₈^{(3/), K}sub 12/, BW_n and Λ₂₄ lattices. Using the proposed method, we have re-derived the trellis diagrams given previously in a systematic and unified approach. It is also shown that for certain complex representation of the Λ₂₄ and the BW_n lattices, we have: (i) the corresponding GS vectors are along the standard co-ordinate system, and (ii) the branch complexity at each section of the resulting trellis meets a given lower bound. This results in a very efficient trellis representation for these lattices