Successive Minimization of the State Complexity of the Self-dual Lattices Using Korkin-Zolotarev Reduced Basis

Authors:
Amir K. Khandani;M. Esmaeili
Affiliations:
Department of Electronics and Computer Engineering, University of Waterloo, Waterloo, ON, Canada, N2L 3G1khandani@shannon.uwaterloo.ca;Department of Electronics and Computer Engineering, University of Waterloo, Waterloo, ON, Canada, N2L 3G1
Venue:
Designs, Codes and Cryptography
Year:
2001

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Abstract

This work presents a systematic method to successively minimize the state complexity of the self-dual lattices (in the sense that each section of the trellis has the minimum possible number of states fixing its preceding co-ordinates). 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. As part of the computations, we give expressions for the GS vectors of a KZ basis of the K_{12}, \Lambda_{24}, and BW_n lattices. It is also shown that for the complex representation of the \Lambda_{24} and the BW_n lattices over the set of the Gaussian integers, 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 certain lower bound. This results in a very efficient trellis representation for these lattices over the standard co-ordinate system.