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A line packing is optimal if its coherence is as small as possible. Mostinteresting examples of optimal line packings are achieving equality in some ofthe known lower bounds for coherence. In this paper two infinite families ofreal and complex biangular line packings are presented. New packings achieveequality in the real or complex second Levenshtein bound respectively. Bothinfinite families are constructed by analyzing well known representations ofthe finite groups SL$(2,\mathbb{F}_q)$. Until now the only known infinitefamiles meeting the second Levenshtein bounds were related to the maximal setsof mutually unbiased bases (MUB). Similarly to the line packings related to themaximal sets of MUBs, the line packings presented here are related to themaximal sets of mutually unbiased weighing matrices. Another similarity is thatthe new packings are projective 2-designs. The latter property together withsufficiently large cardinalities of the new packings implies some improvementon largest known cardinalities of real and complex biangular tight frames.

Infinite families of optimal systems of biangular lines related to representations of $\textrm{SL}(2,\mathbb{F}_q)$