A new class of multiple-rate codes based on block Markov superposition transmission

Hadamard transform (HT) over the binary field provides a natural way to implement multiple-rate codes (referred to as 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 logN). 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. The BMST introduces memory among short HT-coset codes, resulting in long codes. The encoding can be as fast as the short HT-coset codes, while the decoding can be implemented with a sliding-window algorithm. Most importantly, the performance around 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 BMST of short HT-coset codes performs well (within one dB away from the corresponding Shannon limits) in a wide range of code rates.