On RCD codes as a class of LDPC codes: Properties, decoding, and performance evaluation.

摘要

We study a class of regular low-density parity-check (LDPC) codes constructed on eta x eta square arrays with eta prime, with the rows, columns, and diagonals of the array defining the parity-check equations. We denote such codes the row-column-diagonal (RCD) codes.The class of RCD codes are attractive because of their high code rates coupled with the possibility of decoding via a large number of decoding techniques, as per the needs of the application. Practical decoders for RCD codes can be implemented with lower complexity than decoders for most other codes, which makes them desirable candidates for implementation in high-speed applications, e.g., optical fiber communications. RCD codes are also desirable on account of their tractability to combinatorial analysis, thanks to their relatively simple construction, which is helpful in obtaining bounds on RCD code performance in a variety of situations. Availability of these bounds, in turn, makes RCD codes excellent candidates for the testing and validation of techniques that estimate code performance.We show that the general construction technique based on diagonals in eta x eta square arrays with eta prime is equivalent to certain other LDPC code construction techniques based on Euclidean geometries and partial balanced incomplete-block designs. The performance of the RCD codes, which have a minimum distance of 6, is evaluated and analytically bounded (where possible) for several channel assumptions and decoding schemes: (a) the binary symmetric channel (BSC) with decoding via the bit-flipping algorithm (BFA) (b) the binary symmetric channel with erasures (BSC/E), the simplest channel model with soft-information at the channel output, with decoding based on the extended-BFA (e-BFA) and (c) the binary-input, soft-output, additive white Gaussian noise (AWGN) channel using soft-decision decoding via the sum-product algorithm (SPA).We develop two new importance sampling (IS) simulation techniques for evaluating word-error-rate (WER)/bit-error rate (BER) performance of codes to very low values more efficiently that standard Monte Carlo techniques: (a) dual adaptive importance sampling (DAIS) for soft-decision decoding performance, and (b) SNR-invariant importance sampling (IIS) for hard-decision decoding. We provide numerous simulated WER/BER performance results for RCD codes for WERs down to 10-15 or lower, and validate our simulation techniques via comparison with standard Monte Carlo results (at higher WERs) and analytically computed bounds (at lower WERs). These simulation results also serve to demonstrate the behavior of the class of RCD codes and various decoders over a range of SNR values.Finally, we demonstrate that RCD codes and codes based on diagonals in square arrays can serve as useful building blocks for constructing quantum codes that can correct for errors in the quantum domain.