New bounds for equiangular lines and spherical two-distance sets

A set of lines in Rⁿ is called equiangular if the angle between each pair of lines is the same. We derive new upper bounds on the cardinality of equiangular lines. Let us denote the maximum cardinality of equiangular lines in Rⁿ with the common angle arccos α by M_α(n). We prove that M_1/a (n) ≤ 1/2 (a² - 2)(a² - 1) for any n ∈ N in the interval a² - 2 ≤ n ≤ 3a² - 16 and a ≥ 3. Moreover, we discuss the relation between equiangular lines and spherical two-distance sets and we obtain the new results on the maximum spherical two-distance sets in Rⁿ up to n ≤ 417.