Random Matrices from Linear Codes and Wignerxe2x80x99s Semicircle Law II Recently we considered a new normalization of matrices obtained by choosing distinct codewords at random from linear codes over finite fields and proved that under some natural algebraic conditions their empirical spectral distribution converges to Wignerxe2x80x99s semicircle law as the length of the codes goes to infinity. One of the conditions is that the dual distance of the codes is at least 5. In this report, by employing more advanced techniques related to Stieltjes transform, we show that the dual distance being at least 5 is sufficient to ensure the convergence. We also obtain a fast convergence rate in terms of the length of the code.

New Bounds for GLD Lattices and Codes We prove that the ensemble of random Generalized LowDensity (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.