A New Class of Multiple-rate Codes Based on Block Markov Superposition Transmission

TL;DR

This paper introduces a novel approach to enhance short HT-coset codes using block Markov superposition transmission, enabling efficient long code performance with predictable error bounds and near-Shannon limit performance across various rates.

Contribution

It proposes a new method to transmit short HT-coset codes via BMST, improving performance and decoding efficiency while providing simple performance bounds.

Findings

Achieves near-Shannon limit performance within one dB for various code rates.

Provides a simple genie-aided lower bound for BER prediction.

Demonstrates effective long code performance from short HT-coset codes.

Abstract

Hadamard transform~(HT) as over the binary field provides a natural way to implement multiple-rate codes~(referred to as {\em HT-coset codes}), where the code length $N=2^p$ is fixed but the code dimension $K$ can be varied from $1$ to $N-1$ by adjusting the set of frozen bits. The HT-coset codes, including Reed-Muller~(RM) codes and polar codes as typical examples, can share a pair of encoder and decoder with implementation complexity of order $O(N \log N)$. However, to guarantee that all codes with designated rates perform well, HT-coset coding usually requires a sufficiently large code length, which in turn causes difficulties in the determination of which bits are better for being frozen. In this paper, we propose to transmit short HT-coset codes in the so-called block Markov superposition transmission~(BMST) manner. At the transmitter, signals are spatially coupled via superposition, resulting in long codes. At the receiver, these coupled signals are recovered by a sliding-window iterative soft successive cancellation decoding algorithm. Most importantly, the performance around or below the bit-error-rate~(BER) of $10^{-5}$ can be predicted by a simple genie-aided lower bound. Both these bounds and simulation results show that the BMST of short HT-coset codes performs well~(within one dB away from the corresponding Shannon limits) in a wide range of code rates.