$(25)$ A theoretical measure of error$-$correcting capability with the bound$-$distance decoder is Hamming bound which is a different measure from the Hamming bound associated with the sphere packing bound. The Hamming bound is defined as the maximum error rate below which the decoder failure probability of the bounded distance decoding is less than a target rate, which is obtained by increasing the radius of spheres for a given rate. Find the Hamming bound for a code of rate $0\mathrm{.}75$ at length $32,64$ and $128$ when the target decoder failure rate is ${10}^{-2}.$