On Decoding First- and Second-Order BiD Codes

TL;DR

This paper introduces efficient decoding algorithms for first- and second-order BiD codes, which are algebraic codes capable of achieving channel capacity and having larger minimum distances than Reed-Muller codes.

Contribution

It proposes fast ML and max-log-MAP decoders for first-order BiD codes and designs a belief propagation decoder for second-order codes using their parity check structure.

Findings

Fast ML and max-log-MAP decoders for first-order BiD codes.

Belief propagation decoder for second-order codes performs within 1 dB of ML decoder.

BiD codes achieve capacity and have larger minimum distance than Reed-Muller codes.

Abstract

BiD codes, which are a new family of algebraic codes of length $3^m$, achieve the erasure channel capacity under bit-MAP decoding and offer asymptotically larger minimum distance than Reed-Muller (RM) codes. In this paper we propose fast maximum-likelihood (ML) and max-log-MAP decoders for first-order BiD codes. For second-order codes, we identify their minimum-weight parity checks and ascertain a code property known as 'projection' in the RM coding literature. We use these results to design a belief propagation decoder that performs within 1 dB of ML decoder for block lengths 81 and 243.