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
$\mathscr {C}_{n}(r,m)$
, and their duals are denoted as
$\mathscr {B}_{n}(r,m)$
. The length of these codes is
$n^{m}$
, where
$n \geq 2$
, and
$r$
is their ‘order’. When
$n=2$
,
$\mathscr {C}_{n}(r,m)$
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
$\mathscr {B}_{n}(r,m)$
as the Berman code and
$\mathscr {C}_{n}(r,m)$
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, 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
$\mathscr {B}_{n}(r,m)$
and
$\mathscr {C}_{n}(r,m)$
and which achieves BEC capacity.