On the Decoding of Lattices Constructed via a Single Parity Check

Vincent Corlay, Joseph J. Boutros, Philippe Ciblat, Loïc Brunel

Abstract: This paper investigates the decoding of a remarkable set of lattices: We treat in a unified framework the Leech lattice in dimension 24, the Nebe lattice in dimension 72, and the Barnes-Wall lattices. A new interesting lattice, named <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$L_{3\cdot 24}$ </tex-math></inline-formula> , is constructed as a simple application of the single parity check on the Leech lattice. The common aspect of these lattices is that they can be obtained via a single parity check or via the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> -ing construction. We exploit these constructions to introduce a new efficient paradigm for decoding. This leads to efficient list decoders and quasi-optimal decoders on the Gaussian channel. Both theoretical and practical performance (point error probability and complexity) of the new decoders are provided.