Exponential packing of a Hamming cube ($|S|\ge e^{9n/32},d_H(s,s^{\prime })\ge n/8$)

An inclusion-maximal separated set of separation $an$ covers the Hamming cube by strict-radius $an$ balls. The Hoeffding lower-tail bound for Hamming balls then gives at least $e^{2n(1/2-a{)}^2}$ selected vertices. At $a=1/8$ this yields $e^{9n/32}$ vertices, all separated by at least $n/8$.