Hamming ball volume exponent (${\lim }_{n\to \infty }{n}^{-1}{\log }_q{V}_q(n,\lfloor \rho n\rfloor )=H_q(\rho )$)

For $0\le \rho \le 1-1/q$, the exponential growth rate of a q-ary Hamming ball is $H_q(\rho )$. Stirling's formula gives the exponent of each summand $\left(\genfrac{}{}{0ex}{}{n}{i}\right)(q-1{)}^i$, and the sum lies between its largest term and $n+1$ times that term. Above the entropy-maximizing radius fraction, the exponent is one.

// Target: algebra.bigb