More approximately orthogonal vectors than the dimension of the space

This question has been asked and answered on MathOverflow. I have replicated the accepted answer by Lucia below.

A set of points on the unit sphere in ${\Bbb R}^n$ with $\langle x,y\rangle \le \cos \theta$ for all distinct $x$ and $y$ is called a spherical code with minimum angle $\theta$. For $0<\theta < \pi/2$, Kabatiansky and Levenshtein gave an exponential upper bound (of the form $\exp(C(\theta)n)$) for the maximum number of points in such a spherical code. There is also an exponential lower bound. This is related to sphere packings. See for example the recent paper by Cohn and Zhao, which will have more references: http://arxiv.org/pdf/1212.5966v2.pdf