New Bounds for GLD Lattices and Codes

Abstract

We prove that the ensemble of random Generalized Low-Density (GLD) lattices can attain the Poltyrev limit for an alphabet size increasing polylogarithmically with the lattice dimension. Our main theorem imposes no constraints on the normalized minimum distance of the code associated to the lattice ensemble, any asymptotically good code is suitable. This is a great improvement with respect to the first theorem on Poltyrev goodness of GLD lattices (2015). Our new bound is based on a new method referred to as the buckets approach where we employ the asymptotics of the restricted compositions of the Hamming weight. The new bound has applications in many coding areas beyond the specific lattice ensemble considered in this paper.