A generalization of highly symmetric frames is presented by considering also projective stabilizers of frame vectors. This allows construction of highly symmetric line systems and study of highly symmetric frames in a more unified manner. Construction of highly symmetric line systems involves computation of twisted spherical functions associated with finite groups. Further generalizations include definition of highly symmetric systems of subspaces. We give several examples which illustrate our approach including 3 new kissing configurations which improve lower bounds on the kissing number in d = 10, 11, 14 to 510, 592 and 1932 respectively.

Line systems passing through the origin of the d-dimensional Euclidean space admitting exactly two distinct angles are called biangular. It is shown that the maximum cardinality of biangular lines is at least $$2(d-1)(d-2)$$ 2 ( d - 1 ) ( d - 2 ) , and this result is sharp for $$d\in \{4,5,6\}$$ d ∈ { 4 , 5 , 6 } . Connections to binary codes, few-distance sets, and association schemes are explored, along with their multiangular generalization.

Line systems passing through the origin of the d dimensional Euclidean space admitting exactly two distinct angles are called biangular. It is shown that the maximum cardinality of biangular lines is at least 2(d−1)(d−2), and this result is sharp for d ∈ {4, 5, 6}. Connections to binary codes, fewdistance sets, and association schemes are explored, along with their multiangular generalization.