Berman Codes: A Generalization of Reed-Muller Codes that Achieve BEC Capacity

We identify a family of binary codes whose structure is similar to Reed-Muller (RM) codes and which include RM codes as a strict subclass. The codes in this family are denoted as ${\mathcal{C}_n}(r,m)$, and their duals are denoted as ${{\mathcal{B}}_n}(r,m)$. The length of these codes is n
^m
, where n ≥ 2, and r is their ‘order’. When n = 2, ${\mathcal{C}_n}(r,m)$ is the RM code of order r and length 2
^m
. The special case of these codes corresponding to n being an odd prime was studied by Berman (1967) and Blackmore and Norton (2001). Following the terminology introduced by Blackmore and Norton, we refer to ${{\mathcal{B}}_n}(r,m)$ as the Berman code and ${\mathcal{C}_n}(r,m)$ as the dual Berman code. We identify these codes using a recursive Plotkin-like construction, and we show that these codes have a rich automorphism group. Applying a result of Kumar et al. (2016) to this set of automorphisms, we show that these codes achieve the capacity of the binary erasure channel (BEC) under bit-MAP decoding.