Small codes

Abstract

Determining the maximum number of unit vectors in with no pairwise inner product exceeding is a fundamental problem in geometry and coding theory. In 1955, Rankin resolved this problem for all , and in this paper, we show that the maximum is for all , answering a question of Bukh and Cox. Moreover, the exponent is best possible. As a consequence, we obtain an upper bound on the size of a ‐ary code with block length and distance , which is tight up to the multiplicative factor for any prime power and infinitely many . When , this resolves a conjecture of Tietäväinen from 1980 in a strong form and the exponent is best possible. Finally, using a recently discovered connection to ‐ary codes, we obtain analogous results for set‐coloring Ramsey numbers.