New upper bounds on error exponents

We derive new upper bounds on the error exponents for the maximum-likelihood decoding and error detecting in the binary symmetric channels. This is an improvement on the best earlier known bounds by Shannon-Gallager-Berlekamp (1967) and McEliece-Omura (1977). For the probability of undetected error the new bounds are better than the bounds by Levenshtein (1978, 1989) and the bound by Abdel-Ghaffar (see ibid., vol.43, p.1489-502, 1997). Moreover, we further extend the range of rates where the undetected error exponent is known to be exact. The new bounds are based on an analysis of possible distance distributions of the codes along with some inequalities relating the distance distributions to the error probabilities.