The conjecture of Ruján on error-correcting codes is proven. Errors in decoding of signals transmitted through noisy channels assume the smallest values when signals are decoded at a particular finite temperature. This finite-temperature decoding is compared with the conventional maximum likelihood decoding which corresponds to the T =0 case. The method of gauge transformation in the spin glass theory is useful in this argument.
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Hidetoshi Nishimori (1993) studied this question.
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