Polynomial complexity of polar codes for non-binary alphabets, key agreement and Slepian-Wolf coding

Abstract

Abstract-We consider polar codes for memoryless sources with side information and show that the blocklength, construction, encoding and decoding complexities are bounded by a polynomial of the reciprocal of the gap between the compression rate and the conditional entropy. This extends the recent results of Guruswami and Xia to a slightly more general setting, which in turn can be applied to (1) sources with non-binary alphabets, (2) key generation for discrete and Gaussian sources, and (3) SlepianWolf coding and multiple accessing. In each of these cases, the complexity scaling with respect to the number of users is also controlled. In particular, we construct coding schemes for these multi-user information theory problems which achieve optimal rates with an overall polynomial complexity.

An Entropy Sumset Inequality and Polynomially Fast Convergence to Shannon Capacity Over All Alphabets

“…The idea is to use the multi-level code construction technique sketched in [STA09] (and also recently in [LA14] for alphabets of size 2 m ). We outline the procedure here.…”