On the decoding of Barnes-Wall lattices

A Simple Construction for the Barnes-Wall Lattices

A certain family of orthogonal groups (called "Clifford groups" by G. E. Wall) has arisen in a variety of different contexts in recent years. These groups have a simple definition as the automorphism groups of certain generalized Barnes-Wall lattices. This leads to an especially simple construction for the usual Barnes-Wall lattices. This is based on the third author's talk at the Forney-Fest, M.I.T., March 2000, which in turn is based on our paper "The Invariants of the Clifford Groups", Designs, Codes, Crypt., 24 (2001), 99--121, to which the reader is referred for further details and proofs…

Practical Encoders and Decoders for Euclidean Codes from Barnes-Wall Lattices

In this paper, we address the design of high spectral-efficiency Barnes-Wall (BW) lattice codes which are amenable to low-complexity decoding in additive white Gaussian noise (AWGN) channels. We propose a new method of constructing complex BW lattice codes from linear codes over polynomial rings, and show that the proposed construction provides an explicit method of bit-labeling complex BW lattice codes. To decode the code, we adapt the low-complexity sequential BW lattice decoder (SBWD) recently proposed by Micciancio and Nicolosi. First, we study the error performance of SBWD in decoding the…

辛巴恩斯-沃尔GKP码：确定性O(N log²N)译码与对数速率缩放

Symplectic Barnes-Wall GKP Codes: Deterministic $O(N \log^2 N)$ Decoding and Logarithmic Rate Scaling

该研究构造了具有对数编码效率的辛巴恩斯-沃尔GKP码，实现了确定性O(N log²N)译码，为支持非局域模块化连接的平台提供了空间高效的GKP纠错方案。

Barnes-Wall 格 $\Lambda_{16}$ 的 Voronoi 区域

The Voronoi Region of the Barnes-Wall Lattice $Λ_{16}$

本文刻画 Barnes-Wall 格 $\Lambda_{16}$ 的 Voronoi 区域，精确求出其量化常数，并用更准确的新型方差量化方法验证数值估计。

List Decoding Barnes-Wall Lattices

The question of list decoding error-correcting codes over finite fields (under the Hamming metric) has been widely studied in recent years. Motivated by the similar discrete structure of linear codes and point lattices in R^N, and their many shared applications across complexity theory, cryptography, and coding theory, we initiate the study of list decoding for lattices. Namely: for a lattice L in R^N, given a target vector r in R^N and a distance parameter d, output the set of all lattice points w in L that are within distance d of r. In this work we focus on combinatorial and algorithmic que…

On the decoding of lattices constructed via a single parity check

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 Ga…