Small codes
Igor Balla
TL;DR
The paper studies the maximal size of sets of unit vectors in $\mathbb{R}^r$ with bounded pairwise inner products, formalized as $\rho(r,n)$. Building on Rankin's results and recent bounds, it proves $\rho(r,2r+k)=\Theta(1/r)$ for fixed $k$ as $r\to\infty$, and derives sharp consequences for spherical codes, namely $M(r,\alpha)=(2+o(1))r$ when $\alpha=o(r^{-2/3})$. Through a spherical-code to $q$-ary-code reduction, it shows $A_q(r,(1-1/q)r-j)\le (2+o(1))(q-1)r$ for $j=o(r^{1/3})$, tight up to a constant factor, resolving a strong form of Tietäväinen’s conjecture for $q=2$ with the $1/3$ exponent best possible. The work also translates these bounds to set-coloring Ramsey numbers via a recent code-Ramsey connection, highlighting broad implications in geometry, coding theory, and combinatorics.
Abstract
Determining the maximum number of unit vectors in $\mathbb{R}^r$ with no pairwise inner product exceeding $α$ is a fundamental problem in geometry and coding theory. In 1955, Rankin resolved this problem for all $α\leq 0$ and in this paper, we show that the maximum is $(2+o(1))r$ for all $0 \leq α\ll r^{-2/3}$, 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 Tietäväinen 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.