A new framework for building and decoding group codes
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This paper investigates the construction and the decoding of a remarkable set of lattices and codes viewed as group codes: We treat in a unified framework the Leech lattice and the Golay code in dimension 24, the Nebe lattice in dimension 72, the Barnes-Wall lattices, and the Reed-Muller codes. We also present a new family of lattices called the parity lattices. The common aspect of these group codes is that they can be obtained as single parity-check k-groups or via the k-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.
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