A Family of Capacity-Achieving Abelian Codes for the Binary Erasure Channel

We identify a large family of abelian codes that achieve the capacity of the binary erasure channel (BEC) under bit-MAP decoding. The codes in this family have rich automorphism groups, their lengths are odd integers, and they can asymptotically (in the block length) achieve any code rate. This family contains codes of prime power block lengths that were originally identified by Berman (1967) and later inves-tigated by Blackmore and Norton (2001), and also contains their generalization to any odd block length. We use Rajan and Siddiqi's (1992) characterization of abelian codes using discrete Fourier transform (DFT) to identify our code family and study their automorphism groups. We then use a result of Kumar, Calderbank and Pfister (2016) that relates the automorphism group of a code to its performance in the BEC to show that this code family achieves BEC capacity. The full version of this paper including the proofs of all claims and simulation results is available online [1]