Equation 677 Database

Steiner-line magma on a (41, 5, 1)-BIBD with only translation symmetry (Type B slope pattern). Size 41, fully idempotent, RC. 82 size-5 sub-magmas (each = the F_5 affine line magma#e549b5f8), every pair of distinct elements lies in exactly one block. |Aut(M)| = 41 (only translations Z_41, no multiplicative stabilizer). 1 point-orbit; 2 block-orbits of 41 blocks (cyclic BIBD difference family with 2 base blocks). In the suggested reorder, F_41 labeling makes translation x -> x+1 an automorphism. The operation is x*y = x + f(y - x) mod 41 with f a non-linear permutation of F_41. Slope analysis (alpha(y) = f(y)/y on F_41*): Phi_10-root slopes {4, 23, 25, 31} each occur on 6-element subsets (total 24 elements) Eight non-Phi_10 slopes {11, 13, 15, 17, 19, 27, 29, 38} each occur on 2-element subsets (total 16 elements) This "Type B" slope multiset (4*6 + 8*2 = 40) is shared with magma#8b36eeff. So 2 of the 7 translation-only size-41 Steiner-line magmas in the DB have this pattern; the other 5 (magma#d32b07ca, magma#c0b238e3, magma#10305dec, magma#8f879c02, magma#c754a6c0) have a complementary "Type A" pattern (4*2 + 8*4 = 40). Compare with magma#b7e8bf90 (Type "all-Phi": only Phi_10 slopes, each on size-10 set, |Aut| = 205, fully Z_5-symmetric). [text written by Claude]