On short-length error-correcting codes for 5G-NR

We compare the performance of a selection of short-length and very short-length linear binary error-correcting codes on the binary-input Gaussian noise channel, and on the fast and quasi-static flat Rayleigh fading channel. We use the probabilistic Ordered Statistics Decoder, that is universal to any code construction. As such we compare codes and not decoders. The word error rate versus the signal-to-noise ratio is found for LDPC, Reed-Muller, polar, turbo, Golay, random, and BCH codes at length 20, 32 and 256 bits. BCH and random codes outperform other codes in absence of a cyclic redundancy check concatenation. Under joint decoding, the concatenation of a cyclic redundancy check makes all codes perform very close to optimal lower bounds. Optimizations of the Ordered Statistics Decoder are discussed and revealed to bring near-ML performance with a notable complexity reduction, making the decoding complexity at very short length affordable.