Equation 677 Database

Steiner-line magma on a (41, 5, 1)-BIBD with only translation symmetry. Size 41, fully idempotent, RC. Like magma#b7e8bf90, this is a Steiner-line magma: 82 = 41*40/(5*4) size-5 sub-magmas covering each pair exactly once, each block isomorphic to magma#e549b5f8 (the F_5 affine line). But this magma uses a *different* Steiner design - only 11 of the 82 blocks coincide with those of magma#b7e8bf90. |Aut(M)| = 41 (versus 205 for magma#b7e8bf90). The full Aut group is just the cyclic Z_41 of translations; there is no multiplicative stabilizer. As a result, only one Aut orbit (all 41 points), and the 82 blocks split as exactly 2 Aut-orbits of 41 blocks each (the cyclic BIBD's 2 base-block translate classes). In the suggested reorder, elements are labeled by F_41 so that translation x -> x+1 is an automorphism. The operation is x*y = x + f(y - x) mod 41, where f is a non-linear permutation of F_41 with f(0) = 0. Slope analysis: writing f(y) = alpha(y) * y, the slope function takes 12 distinct values: Phi_10-root slopes (the four primitive 10th roots of unity in F_41*): {4, 23, 25, 31}, each occurring on a size-2 subset alpha=4 on {21, 31} alpha=23 on {35, 38} alpha=25 on {13, 26} alpha=31 on {1, 2} Non-Phi_10 slopes (8 values), each occurring on a size-4 subset: alpha=11 on {4, 8, 9, 25} alpha=13 on {10, 20, 33, 37} alpha=15 on {3, 6, 17, 29} alpha=17 on {11, 12, 22, 24} alpha=19 on {7, 14, 19, 30} alpha=27 on {15, 27, 28, 34} alpha=29 on {5, 23, 39, 40} alpha=38 on {16, 18, 32, 36} Totals: 4*2 + 8*4 = 40 = |F_41*|. So 8 elements get a Phi_10-root slope; the other 32 use non-root slopes - meaning f is not a "linear-on-cosets" function as in magma#b7e8bf90. It's a genuinely Tao-Type-II non-linear permutation of F_41 satisfying Eq 677 only because of compatibility conditions on the f values across blocks. The four Phi_10-root slopes appearing on 2-element subsets is a striking pattern - it suggests a partial structure where some pairs of differences are "fully aligned" with the abstract block magma, while most differences carry a separate slope. Comparison within size 41: magma#345a4ec9, magma#385ad4434a17, magma#a7a4546c, magma#ef76453c -- the four F_41 affine line magmas (one per Phi_10 root, single-slope linear over F_41), |Aut| = 1640 = 40 * 41. magma#b7e8bf90 -- Tao Type II piecewise with 4 Phi_10-root slopes on Z_5-cosets, |Aut| = 205. this magma#d32b07ca -- Tao Type II with 12 slopes, |Aut| = 41 (only translation). [text written by Claude]