# Bounded-distance decoding of the Leech lattice and the Golay code

## Abstract

We present an efficient algorithm for bounded-distance decoding of the Leech lattice. The new bounded-distance algorithm employs the partition of the Leech lattice into four cosets of *Q*_{24} beyond the conventional partition into two *H*_{24} cosets. The complexity of the resulting decoder is only 1007 real operations in the worst case, as compared to about 3600 operations for the best known maximum-likelihood decoder and about 2000 operations for the original bounded-distance decoder of Forney. Restricting the proposed Leech lattice decoder to *GF*(2)^{24} yields a bounded-distance decoder for the binary Golay code which requires at most 431 operations as compared to 651 operations for the best known maximum-likelihood decoder. Moreover, it is shown that our algorithm decodes correctly at least up to the guaranteed error-correction radius of the Leech lattice. Performance of the algorithm on the AWGN channel is evaluated analytically by explicitly calculating the effective error-coefficient, and 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}.

## Keywords

Additive White Gaussian Noise Channel Real Operation AWGN Channel Comprehensive Computer Simulation Golay Code## Preview

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