Spherical codes with minimal angle arccos(1/3) in dimensions four, five and six

Let A(n, θ) be the largest number of points on the unit sphere in R^n with pairwise angular distance at least θ. We prove A(4, arccos 1/3) = 14, A(5, arccos 1/3) ≤ 22 and A(6, arccos 1/3) ≤ 34. The previous upper bounds were 15, 23 and 35. For n = 5 the best known code has 21 points, so A(5, arccos 1/3) is 21 or 22. Each upper bound comes from an exact rational solution of the second level of the Lasserre hierarchy for spherical codes, in the formulation of de Laat, Leijenhorst and de Muinck Keizer, and was checked with their verification program. The largest number of points at angle π/3 from a given point of a kissing configuration in R^n equals A(n-1, arccos 1/3). So this number is 14 in R^5, 21 or 22 in R^6, and at most 34 in R^7.