Number of words within Hamming distance $\delta$
That's an inaccuracy. If $\delta &lt; 1/2$ is constant then it is the case that $\sum_{k=0}^{\delta m} \binom{m}{k} = O\left(\binom{m}{\delta m}\right)$, since the binomial coefficients increase very fast until they finally level out around the central binomial coefficient. The hidden constant depends on $\delta$. (When $\delta$ is close to $1/2$ this doesn't hold: for example, $\sum_{k=0}^{m/2} \binom{m}{k} = 2^{m-1}$ for even $m$, while $\binom{m}{m/2} = \Theta\left(\frac{2^m}{\sqrt{m}}\right)$.)

At any rate, what they are interested in really is the estimate $\sum_{k=0}^{\delta m} \binom{m}{k} \leq (1-\epsilon) 2^{H(\delta)m}$, which holds for every $\epsilon &gt; 0$ and large enough $m$ (depending on $\epsilon,\delta$).