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.

Asymptotically good ZprZpsadditive cyclic codes Abstract   We construct a class of      Z      p    r        Z      p    s       additive cyclic codes generated by pairs of polynomials, where p is a prime number. Based on probabilistic arguments, we determine the asymptotic rates and relative distances of this class of codes: the asymptotic GilbertVarshamov bound at      1      p    s  xe2x88x92  r      2    xcexb4    is greater than      1    2      and the relative distance of the code is convergent to xcexb4, while the rate is convergent to      1    1      p    s  xe2x88x92  r        for    0    xcexb4      1    1      p    s  xe2x88x92  r        and    1  xe2x89xa4  r    s   . As a consequence, we prove that there exist numerous asymptotically good      Z      p    r        Z      p    s       additive cyclic codes.