There are no 76 equiangular lines in $R^{19}$

Abstract

Maximum size of equiangular lines in R 19 has been known in the range between 72 to 76 since 1973. Acoording to the nonexistence of strongly regular graph (75, 32,10,16) [1], Larmen-Rogers-Seidel Theorem [20] and Lemmen-Seidel bounds on equiangular lines with common angle 1 3 [22], we can prove that there are no 76 equiangular lines in R 19 . As a corollary, there is no strongly regular graph (76, 35,18,14). Similar discussion can prove that there are no 96 equiangular lines in R 20 .

Equiangular Tight Frames From Hyperovals

“…Further recall that for any such q, the canonical projective plane of order q contains the hyperoval (14). In this case, we construct the requisite unimodular simplex and cosimplex from Hadamard matrices [33]. To increase readability, we denote the rows of a 5 × 6 and 5 × 4 unimodular simplex and cosimplex by the first 10 letters of the alphabet.…”

“…For any q = 2 e where e > 1, the ETF ( 20) is new [11]. Indeed, strongly regular graphs corresponding to real ETFs with these parameters are not known to exist [5], [6], and have been shown to not exist when q = 4 [1], [33]. Moreover, no ETF with these parameters can be harmonic since…”

“…The answer to Problem 2 is "yes" when q = 2, e.g. (4), and "no" when q = 4, since real ETFs with (d, n) = (19, 76) do not exist [1], [33]. To our knowledge, the q = 6 case is open.…”

“…Comparing against known constructions [11], it appears that with the exception of e = 1 case, no real or complex ETF with parameters (3) has been discovered before. In fact, taking e = 2 in (3) gives d = 19 and n = 76, and the existence of a real ETF of this size appears to have been recently ruled out with a computer-assisted search [1], resolving a longstanding open problem [33].…”

Polyphase equiangular tight frames and abelian generalized quadrangles

“…Until recently, the existence of real ETFs with (d, n) parameters (19,76) and (20,96) were longstanding open problems. Both have now been ruled out with computer-assisted arguments [1,2,47]. Notably, [20] shows how to construct complex ETFs of these sizes.…”