  We study a discrete model of repelling particles, and we show using linear
programming bounds that many familiar families of error-correcting codes
minimize a broad class of potential energies when compared with all other codes
of the same size and block length. Examples of these universally optimal codes
include Hamming, Golay, and Reed-Solomon codes, among many others, and this
helps explain their robustness as the channel model varies. Universal
optimality of these codes is equivalent to minimality of their binomial
moments, which has been proved in many cases by Ashikhmin and Barg. We
highlight connections with mathematical physics and the analogy between these
results and previous work by Cohn and Kumar in the continuous setting, and we
develop a framework for optimizing the linear programming bounds. Furthermore,
we show that if these bounds prove a code is universally optimal, then the code
remains universally optimal even if one codeword is removed.
