We show that for a fixed $q$, the number of $q$-ary $t$-error correcting codes of length $n$ is at most $2^{(1 + o(1)) H_q(n,t)}$ for all $t \leq (1 - q^{-1})n - C_q\sqrt{n \log n}$ (for sufficiently large constant $C_q$), 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 $t = o(n^{1/3} (\log n)^{-2/3})$.

翻译：我们显示,对于固定的美元,以美元为单位的折价折价折价折价折价1美元+美元(1美元+美元(1))H_q(n,t)}$(美元+美元1美元)n-C_qqqrt{n\log n}$(美元相当大,不变为C_q美元),其中美元=qqq(n,t)=qn/V_q(n,t)美元为Hamming绑定值和$V_q(n,t)美元为Hamming球半径的基数。这证明了Balogh、Treglown和Wagner的直线值,后者显示了美元=o(n)1/3}(gn,n)%2/3美元的结果。