The Leech lattice, the octacode, and decoding algorithms

New multilevel constructions of the Golay code and the Leech lattice are presented. These are derived from the Turyn construction and the “holy construction” with the octacode as the glue code. Further, we show that the “holy construction” of the Leech lattice with the octacode as the glue code is essentially different from the permuted Turyn construction, although both constructions rely on the octacode. The Turyn construction is based on an “odd” type of the octacode, whereas any type of the octacode can be used in the “holy construction.” Moreover, the multilevel representation of the “holy construction” leads to a novel lattice partition chain. Based on these structures, we derive new bounded-distance decoders for the Golay code and the Leech lattice whose effective error coefficient is smaller than that of any previously known bounded-distance decoder. We provide a general theorem for computing the effective error coefficient of coset decoding with bounded distance decoding for the subcode