Statistical Mechanics and Capacity-Approaching Error-Correcting Codes

Abstract

I will show that there is a deep relation between error-correction codes and certain mathematical models of spin glasses. In particular minimum error probability decoding is equivalent to finding the ground state of the corresponding spin system. The most probable value of a symbol is related to the magnetization at a different temperature. Convolutional codes correspond to one-dimensional spin systems and Viterbi's decoding algorithm to the transfer matrix algorithm of Statistical Mechanics. I will also show how the recently discovered (or rediscovered) capacity approaching codes (turbo codes and low density parity check codes) can be analysed using statistical mechanics. It is possible to show, using statistical mechanics, that these codes allow error-free communication for signal to noise ratio above a certain threshold. This threshold depends on the particular code, and can be computed analytically in many cases.