Soft decoding techniques for codes and lattices, including the Golay code and the Leech lattice

Two kinds of algorithms are considered. <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1)</tex> If *** is a binary code of length <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> , a "soft decision" decoding algorithm for *** changes an arbitrary point of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Rⁿ</tex> into a nearest codeword (nearest in Euclidean distance). <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2)</tex> Similarly, a decoding algorithm for a lattice <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Λ</tex> in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Rⁿ</tex> changes an arbitrary point of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Rⁿ</tex> into a closest lattice point. Some general methods are given for constructing such algorithms, ami are used to obtain new and faster decoding algorithms for the Gosset lattice <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E₈</tex> , the Golay code the Leech lattice.