Result: 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 $$A_w$$ A w codewords of weight w. The parameters of these designs are $$\bigg (A_w,v,w,\frac{wA_w}{v},\lambda _1,\lambda _2\bigg ).$$ ( A w , v , w , w A w v , λ 1 , λ 2 ) . When $$\lambda _1=\lambda _2=\lambda ,$$ λ 1 = λ 2 = λ , the PBIBD becomes a 2- $$(v,w,\lambda )$$ ( v , w , λ ) design. We obtain computationally 111 such designs when G ranges over $$\textrm{L}_2(8){:}3, \textrm{U}_{4}(2), \textrm{U}_{3}(3){:}2, \textrm{A}_8, \textrm{S}_6(2),$$ L 2 ( 8 ) : 3 , U 4 ( 2 ) , U 3 ( 3 ) : 2 , A 8 , S 6 ( 2 ) , $$\textrm{S}_{4}(4), \textrm{U}_{5}(2), \textrm{M}_{11}, \textrm{M}_{22}, \textrm{HS}, \textrm{G}_2(4), \textrm{S}_{8}(2),\textrm{O}^{+}_{10}(2),$$ S 4 ( 4 ) , U 5 ( 2 ) , M 11 , M 22 , HS , G 2 ( 4 ) , S 8 ( 2 ) , O 10 + ( 2 ) , and $$\textrm{O}^{-}_{10}(2)$$ 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), $$28,\, 36$$ 28 , 36 (2), $$\,56,\, 176$$ 56 , 176 points respectively. The design on 176 points is invariant under the Higman–Sims group.