  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.
