Small codes.

Determining the maximum number of unit vectors in Rr$\mathbb {R}^r$ with no pairwise inner product exceeding α$\alpha$ is a fundamental problem in geometry and coding theory. In 1955, Rankin resolved this problem for all α⩽0$\alpha \leqslant 0$, and in this paper, we show that the maximum is (2+o(1))r$(2+o(1))r$ for all 0⩽α≪r−2/3$0 \leqslant \alpha \ll r^{-2/3}$, answering a question of Bukh and Cox. Moreover, the exponent −2/3$-2/3$ is best possible. As a consequence, we obtain an upper bound on the size of a q$q$‐ary code with block length r$r$ and distance (1−1/q)r−o(r1/3)$(1-1/q)r - o(r^{1/3})$, which is tight up to the multiplicative factor 2(1−1/q)+o(1)$2(1 - 1/q) + o(1)$ for any prime power q$q$ and infinitely many r$r$. When q=2$q = 2$, this resolves a conjecture of Tietäväinen from 1980 in a strong form and the exponent 1/3$1/3$ is best possible. Finally, using a recently discovered connection to q$q$‐ary codes, we obtain analogous results for set‐coloring Ramsey numbers. [ABSTRACT FROM AUTHOR]