  Determining the maximum number of unit vectors in $\mathbb{R}^r$ with no
pairwise inner product exceeding $\alpha$ is a fundamental problem in geometry
and coding theory. In 1955, Rankin resolved this problem for all $\alpha \leq
0$ and in this paper, we show that the maximum is $(2+o(1))r$ for all $0 \leq
\alpha \ll r^{-2/3}$, answering a question of Bukh and Cox. Moreover, the
exponent $-2/3$ is best possible. As a consequence, we conclude that when $j
\ll r^{1/3}$, a $q$-ary code with block length $r$ and distance $(1-1/q)r - j$
has size at most $(2 + o(1))(q-1)r$, which is tight up to the multiplicative
factor $2(1 - 1/q) + o(1)$ for any prime power $q$ and infinitely many $r$.
When $q = 2$, this resolves a conjecture of Tiet\"av\"ainen from 1980 in a
strong form and the exponent $1/3$ is best possible. Finally, using a recently
discovered connection to $q$-ary codes, we obtain analogous results for
set-coloring Ramsey numbers.
