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 < f(p, n, d)
, 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.