On noise stability of a system with error-correcting codes

The problem posed in this paper is to give a relation connecting the noise stability of a communication system in which error-correcting codes are used to the parameters of these codes. The amount x of code combinations is found, which differ from each other by not less than a given number of elements for a given arrangement of all the possible combinations. It is proved that the quantity x depends on the arrangement of the primary combinations. Inequalities are obtained for the greatest amount of combinations x_m , for which any two differ by not less than d elements for small and large values n of the common number of elements in each code combination. It is established that the probability of distortion, p_n , of a code group in a system with correcting codes satisfies the inequality p_n , where p is the probability of distortion of one element. The shape of the f(p, n, d) function is found for large values of n . Graphs of this function are constructed.