Primitive rank 3 groups, binary codes, and 3-designs

Let G be a primitive rank 3 permutation group acting on a set of size v. Binary codes of length v globally invariant under G are well-known to hold PBIBDs in their \(A_w\) codewords of weight w. The parameters of these designs are \(\bigg (A_w,v,w,\frac{wA_w}{v},\lambda _1,\lambda _2\bigg ).\) When \(\lambda _1=\lambda _2=\lambda ,\) the PBIBD becomes a 2-\((v,w,\lambda )\) design. We obtain computationally 111 such designs when G ranges over \(\textrm{L}_2(8){:}3, \textrm{U}_{4}(2), \textrm{U}_{3}(3){:}2, \textrm{A}_8, \textrm{S}_6(2),\) \(\textrm{S}_{4}(4), \textrm{U}_{5}(2), \textrm{M}_{11}, \textrm{M}_{22}, \textrm{HS}, \textrm{G}_2(4), \textrm{S}_{8}(2),\textrm{O}^{+}_{10}(2),\) and \(\textrm{O}^{-}_{10}(2)\) in the notation of the Atlas. Included in the counting are 2-designs which are held by nonzero weight codewords of the binary adjacency codes of the triangular and square lattice graphs, respectively. The 2-designs in this paper can be obtained neither from Assmus–Mattson theorem, nor by the classical 2-tra nsitivity (or 2-homogeneity) argument of the automorphism group of the code. Further, the extensions of the codes that hold 2-designs sometimes hold 3-designs. We thus obtain nine self-complementary 3-designs on 16 (4), \(28,\, 36\) (2), \(\,56,\, 176\) points respectively. The design on 176 points is invariant under the Higman–Sims group.