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.

This work together with an independent work by Feng-wen Sun and Henk van-Tilborg will appear in the Information Theory Trans. as a joint paper entitled: “The Leech lattice and the Golay code: Bounded distance decoding and multilevel constructions.”

Research supported in part by the Eshkol Fellowship administered by the Israel Ministry of Science and in part by the Rothschild Fellowship administered by the Rothschild Yad Hanadiv Foundation.