Order 189, glued along a design with a hole: A(37;26,24,0) on the hole, A(5;2,4,0) on every block of five (its points in the order listed), and a point off the hole squares to itself. Two distinct points not both in the hole multiply in their block; two points of the hole, in A(37;26,24,0): x*y = 26x + 24y mod 37, on the labels 0..36. It satisfies Equation 677 because the blocks and the hole do.
The design: a (189, {5, 37*})-PBD, the hole 0..36 and blocks of five meeting it at most once, every pair not both in the hole in exactly one block. The point (x, r), x in Z/19, r in 0..7, is 37 + 19r + x. Blocks off the hole: the translates (x, r) -> (x + t, r) of {(0,0), (1,0), (7,0), (11,0), (0,7)} {(4,0), (9,0), (6,0), (0,1), (0,7)} {(0,1), (13,1), (15,1), (10,1), (0,5)} {(2,1), (14,1), (3,1), (0,3), (0,6)} {(0,2), (16,2), (17,2), (5,2), (0,6)} {(8,2), (18,2), (12,2), (0,0), (0,5)} {(0,3), (2,3), (14,3), (3,3), (0,4)} {(8,3), (18,3), (12,3), (0,2), (0,4)} {(0,4), (8,4), (18,4), (12,4), (0,0)} {(13,4), (15,4), (10,4), (0,3), (0,5)} {(0,5), (8,5), (18,5), (12,5), (0,2)} {(4,5), (9,5), (6,5), (0,2), (0,7)} {(0,6), (2,6), (14,6), (3,6), (0,1)} {(1,6), (7,6), (11,6), (0,1), (0,4)} {(0,7), (13,7), (15,7), (10,7), (0,3)} {(16,7), (17,7), (5,7), (0,0), (0,6)}. A hole point h has a functional f on F_2^3 (r read in binary) and a in (Z/19)^8; its blocks are {h} + {(a_r + t, r) : f.r = e}, listed h first and then by increasing r, for e = 0, 1 and every t. Hole point 0 has the first (f, a) below; each further (f, a) gives the next three hole points, with a, 7a, 11a, in that order: f = 4, a = (0,0,0,0,0,0,0,0); f = 7, a = (0,0,17,16,12,10,3,18); f = 7, a = (0,9,15,7,4,14,17,12); f = 4, a = (0,11,10,13,15,8,0,11); f = 5, a = (0,17,4,13,15,9,6,6); f = 2, a = (0,18,10,6,14,11,8,14); f = 1, a = (0,18,17,5,5,2,10,14); f = 1, a = (0,16,3,8,7,12,9,13); f = 7, a = (0,18,14,2,10,16,7,5); f = 6, a = (0,2,3,0,8,18,12,3); f = 4, a = (0,9,1,8,8,9,7,15); f = 3, a = (0,4,7,9,15,2,8,1); f = 2, a = (0,16,10,14,6,8,6,9).
Equation 255: satisfied. Idempotent elements: 153.
qawbecrdtey · 2026-10-08 15:36:18