The Voronoi Spherical CDF for Lattices and Linear Codes: New Bounds for Quantization and Coding

Abstract: For a lattice/linear code, we define the Voronoi spherical cumulative density function (CDF) as the CDF of the $\ell_2$-norm/Hamming weight of a random vector uniformly distributed over the Voronoi cell. Using the first moment method together with a simple application of Jensen's inequality, we develop lower bounds on the expected Voronoi spherical CDF of a random lattice/linear code. Our bounds are quite close to a trivial ball-based lower bound and immediately translate to improved upper bounds on the normalized second moment and the error probability of a random lattice over the additive white Gaussian noise channel, as well as improved upper bounds on the Hamming distortion and the error probability of a random linear code over the binary symmetric channel.