Spin Glasses, Error-Correcting Codes and Finite-Temperature Decoding

The probability for a sequence of information symbols to have been sent, when the transmission channel's output is known, is simply related to a spin glass Hamiltonian. The ground state of this Hamiltonian provides the most probable input sequence, given the channel's output. Furthermore, the local magnetization at a certain value of the temperature is related to the most probable value of the corresponding information symbol, i.e. finite-temperature decoding provides the sequence of the most probable symbols, which, in general, is different from the most probable sequence.