The Leech lattice and the Golay code: bounded-distance decoding and multilevel constructions

Multilevel constructions of the binary Golay code and the Leech lattice are described. Both constructions are based upon the projection of the Golay code and the Leech lattice onto the (6,3,4) hexacode over GF(4). However, unlike the previously reported constructions, the new multilevel constructions make the three levels independent by way of using a different set of coset representatives for one of the quaternary coordinates. Based upon the multilevel structure of the Golay code and the Leech lattice, efficient bounded-distance decoding algorithms are devised. The bounded-distance decoder for the binary Golay code requires at most 431 operations. As compared to 651 operations for the best known maximum-likelihood decoder. Efficient bounded-distance decoding of the Leech lattice is achieved by means of partitioning it into four cosets of Q₂₄, beyond the conventional partition into two H₂₄ cosets. The complexity of the resulting decoder is only 953 real operations on the average and 1007 operations in the worst case, as compared to about 3600 operations for the best known in maximum-likelihood decoder. It is shown that the proposed algorithms decode correctly at least up to the guaranteed error-correction radius of the maximum-likelihood decoder. Thus, the loss in coding-gain is due primarily to an increase in the effective error-coefficient, which is calculated exactly for both algorithms. Furthermore, the performance of the Leech lattice decoder on the AWGN channel is evaluated experimentally by means of a comprehensive computer simulation. The results show a loss in coding-gain of less than 0.1 dB relative to the maximum-likelihood decoder for BER ranging from 10^-1 to 10 ^-7