A Comparison of Known Codes, Random Codes, and the Best Codes

This paper calculates new bounds on the size of the performance gap between random codes and the best possible codes. The first result shows that, for large block sizes, the ratio of the error probability of a random code to the sphere-packing lower bound on the error probability of every code on the binary symmetric channel (BSC) is small for a wide range of useful crossover probabilities. Thus even far from capacity, random codes have nearly the same error performance as the best possible long codes. The paper also demonstrates that a small reduction k ~ k in the number of information bits conveyed by a codeword will make the error performance of an (n; ~ k) random code better than the sphere-packing lower bound for an (n; k) code as long as the channel crossover probability is somewhat greater than a critical probability. For example, the sphere-packing lower bound for a long (n; k), rate 1=2, code will exceed the error probability of an (n; ~ k) random code if k ~ k > 10 and the crossover probability is between 0:035 and 0:11 = H (1=2). Analogous results are presented for the binary erasure channel (BEC) and the additive white Gaussian noise (AWGN) channel. The paper also presents substantial numerical evaluation of the performance of random codes and existing standard lower bounds for the BEC, BSC, and the AWGN channel. These last results provide a useful standard against which to measure many popular codes including turbo codes, e.g., there exist turbo codes that perform within 0.6 dB of the bounds over a wide range of block lengths.