  We derive a procedure for computing an upper bound on the number of
equiangular lines in various Euclidean vector spaces by generalizing the
classical pillar decomposition developed by (Lemmens and Seidel, 1973); namely,
we use linear algebra and combinatorial arguments to bound the number of
vectors within an equiangular set which have inner products of certain signs
with a negative clique. After projection and rescaling, such sets are also
certain spherical two-distance sets, and semidefinite programming techniques
may be used to bound the size. Applying our method, we prove new relative
bounds for the angle arccos(1/5). Experiments show that our relative bounds for
all possible angles are considerably less than the known SDP bounds for a range
of larger dimension r. Our computational results also show an explicit bound on
the size of a set of equiangular lines regardless of angle, which is strictly
less than the well-known Gerzon's bound if r+2 is not a square of an odd
number.
