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

The 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 the same fundamental encoder/decoder architecture with the implementation complexity of order O(NlogN). 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.