Designing Voronoi Constellations to Minimize Bit Error Rate

In a classical 1983 paper, Conway and Sloane presented fast encoding and decoding algorithms for a special case of Voronoi constellations (VCs), for which the shaping lattice is a scaled copy of the coding lattice. Feng generalized their encoding and decoding methods to arbitrary VCs. Less general algorithms were also proposed by Kurkoski and Ferdinand, respectively, for VCs with some constraints on their coding and shaping lattices. In this work, we design VCs with a cubic coding lattice based on Kurkoski\u27s encoding and decoding algorithms. The designed VCs achieve up to 1.03 dB shaping gains with a lower complexity than Conway and Sloane\u27s scaled VCs. To minimize the bit error rate (BER), pseudo-Gray labeling of constellation points is applied. In uncoded systems, the designed VCs reduce the required SNR by up to 1.1 dB at the same BER, compared with the same VCs using Feng\u27s and Ferdinand\u27s algorithms. In coded systems, the designed VCs are able to achieve lower BER than the scaled VCs at the same SNR. In addition, a Gray penalty estimation method for such VCs of very large size is introduced

Finding a closest point in a lattice of Voronoi's first kind

We show that for those lattices of Voronoi's first kind with known obtuse superbasis, a closest lattice point can be computed in $O(n^4)$ operations where $n$ is the dimension of the lattice. To achieve this a series of relevant lattice vectors that converges to a closest lattice point is found. We show that the series converges after at most $n$ terms. Each vector in the series can be efficiently computed in $O(n^3)$ operations using an algorithm to compute a minimum cut in an undirected flow network

Better Lattice Quantizers Constructed from Complex Integers

Real-valued lattices and complex-valued lattices are mutually convertible, thus we can take advantages of algebraic integers to defined good lattice quantizers in the real-valued domain. In this paper, we adopt complex integers to define generalized checkerboard lattices, especially $\mathcal{E}_{m}$ and $\mathcal{E}_{m}^+$ defined by Eisenstein integers. Using $\mathcal{E}_{m}^+$, we report the best lattice quantizers in dimensions $14$, $18$, $20$, and $22$. Their product lattices with integers $\mathbb{Z}$ also yield better quantizers in dimensions $15$, $19$, $21$, and $23$. The Conway-Sloane type fast decoding algorithms for $\mathcal{E}_{m}$ and $\mathcal{E}_{m}^+$ are given.Comment: 7 page