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) and which include RM as strict subclass. The in this are denoted Cn ( xmlns:xlink="http://www.w3.org/1999/xlink">r,m ), their duals xmlns:xlink="http://www.w3.org/1999/xlink">Bn ). length these xmlns:xlink="http://www.w3.org/1999/xlink">nm , where xmlns:xlink="http://www.w3.org/1999/xlink">n ≥ 2, xmlns:xlink="http://www.w3.org/1999/xlink">r ‘order’. When = ) the code order 2 m . special case corresponding being an odd prime was studied by Berman (1967) Blackmore Norton (2001). Following terminology introduced Norton, we refer xmlns:xlink="http://www.w3.org/1999/xlink">Berman code xmlns:xlink="http://www.w3.org/1999/xlink">dual Berman code. using recursive Plotkin-like construction, show that have rich automorphism group, they generated minimum weight codewords, can be decoded up half distance efficiently. Using result Kumar et al. (2016), achieve capacity erasure channel (BEC) under bit-MAP decoding. Furthermore, except double transitivity, satisfy all properties used Reeves Pfister binary-input memoryless symmetric channels. Finally, when odd, large class abelian includes achieves BEC capacity.