The cardinality of a Hamming ball is the total number of elements that are any distance up to and including its radius.

The number of vectors in Hamming ball $B_{r} (x)$ is

(\binom{n}{0}) + (\binom{n}{1}) + (\binom{n}{2}) + \dots + (\binom{n}{r})

Proof

Recall the definition for $B_{r} (x)$ :

B_{r} (x) = {y \in Z_{2}^{n} : d_{H} (x, y) \leq r}

N.b. $d_{H} (x, y)$ on a binary code in the Hamming space $Z_{2}^{n}$ can only be an integer $0 \leq d_{H} (x, y) \leq n$ .

By Hamming distance for binary codes can be calculated using the binomial coefficient.,
$(\binom{n}{0})$ is the number of elements of $Z_{2}^{n}$ that are distance $0$ from $n$
$(\binom{n}{1})$ is the number of elements of $Z_{2}^{n}$ that are distance $1$ from $n$
$⋮$
$(\binom{n}{t})$ is the number of elements of $Z_{2}^{n}$ that are distance $t$ from $n$

QED