A construction of big convolutional codes from short codes called block Markov superposition transmission (BMST) is proposed. The BMST is very similar to superposition block Markov encoding (SBME), which has been widely used to prove multiuser coding theorems. The BMST codes can also be viewed as a class of spatially coupled codes, where the generator matrices of the involved short codes (referred to as basic codes) are coupled. The encoding process of BMST can be as fast as that of the basic code, while the decoding process can be implemented as an iterative sliding-window decoding algorithm with a tunable delay. More importantly, the performance of BMST can be simply lower bounded in terms of the transmission memory given that the performance of the short code is available. Numerical results show that: 1) the lower bounds can be matched with a moderate decoding delay in the low bit-error-rate (BER) region, implying that the iterative sliding-window decoding algorithm is near optimal; 2) BMST with repetition codes and single parity-check codes can approach the Shannon limit within 0.5 dB at the BER of 10 −5 for a wide range of code rates; and 3) BMST can also be applied to nonlinear codes.
For any given short code (referred to as the basic code), block Markov superposition transmission (BMST) provides a simple way to obtain predictable extra coding gain by spatially coupling the generator matrix of the basic code. This paper presents a systematic design methodology for BMST systems to approach the channel capacity at any given target bit error rate (BER) of interest. To simplify the design, we choose the basic code as the Cartesian product of a short block code. The encoding memory is then inferred from the genie-aided lower bound according to the performance gap of the short block code to the corresponding Shannon limit at the target BER. In addition to the sliding-window decoding algorithm, we propose to perform one more phase decoding to remove residual (rare) errors. A new technique that assumes a noisy genie is proposed to upper bound the performance. Under some mild assumptions, these genie-aided bounds can be used to predict the performance of the proposed two-phase decoding algorithm in the extremely low BER region. Using the Cartesian product of a repetition code as the basic code, we construct a BMST system with an encoding memory 30 whose performance at the BER of 10 −15 can be predicted within 1 dB away from the Shannon limit over the binary-input additive white Gaussian noise channel.Index Terms-Big convolutional codes, block Markov superposition transmission (BMST), capacity-approaching codes, genieaided bounds, spatial coupling, two-phase decoding (TPD).
Hadamard transform (HT) as 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 ReedMuller (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.
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