On the decoding of lattices constructed via a single parity check

TL;DR

This paper presents a unified decoding framework for special lattices like the Leech, Nebe, and Barnes-Wall lattices, utilizing single parity-check principles to achieve efficient list and quasi-optimal decoding on Gaussian channels.

Contribution

It introduces a novel decoding paradigm based on single parity-check and k-ing constructions, applicable to multiple notable lattices, improving decoding efficiency and performance.

Findings

Efficient list decoders developed for the lattices.

Decoders achieve quasi-optimal performance on Gaussian channels.

Theoretical and practical analysis confirms improved error probability and complexity.

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 is constructed as a simple application of single parity-check principle on the Leech lattice. The common aspect of these lattices is that they can be obtained via a single parity check 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.