Abstract
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 Cn(r, m), and their duals are denoted as Bn(r, m). The length of these codes is n m, where n ≥ 2, and r is their ‘order’. When n = 2, Cn(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 Bn(r, m) as the Berman code and Cn(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, they are generated by the minimum weight codewords, and that they can be decoded up to half the minimum distance efficiently. Using a result of Kumar et al. (2016), we show that these codes achieve the capacity of the binary erasure channel (BEC) under bit-MAP decoding. Furthermore, except double transitivity, they satisfy all the code properties used by Reeves and Pfister to show that RM codes achieve the capacity of binary-input memoryless symmetric channels. Finally, when n is odd, we identify a large class of abelian codes that includes Bn(r, m) and Cn(r, m) and which achieves BEC capacity
