On the number of error correcting codes

We show that for a fixed q, the number of q-ary t-error correcting codes of length n is at most 2((1+o(1))Hq(n,t)) for all t <= (1 - q(-1))n - 2 root n log n, where H-q(n, t) = q(n)/V-q(n, t) is the Hamming bound and V-q(n, t) is the cardinality of the radius t Hamming ball. This proves a conjecture of Balogh, Treglown, and Wagner, who showed the result for t = o(n(1/3)(log n)(-2/3)).