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.

翻译：BiD码是一类长度为$3^m$的新型代数码，在比特最大后验概率译码下可实现擦除信道的容量，且其渐近最小距离大于Reed-Muller码。本文针对一阶BiD码提出了快速最大似然译码器与最大对数后验概率译码器。对于二阶码，我们确定了其最小权重校验子，并验证了Reed-Muller编码理论中称为“投影”的码性质。基于这些结果，我们设计了置信传播译码器，在码长为81与243时，其性能与最大似然译码器相差在1 dB以内。