Non-symmetric nearly triply regular designs Original Research Article

A non-symmetric 2-(v, k, λ) design D is said to be nearly triply regular (NTR) if there are positive integers u, vl and v2 with v1 > v2 such that (i) |α ∩ β| is 0 orΠ, for any two distinct blocks of D (that is, D is quasi-symmetric) and (ii) |α∩β∩γ| is 0, v1, or v2, for any three distinct blocks α,β,γ of D. If these conditions hold with v1 = v2, we say that D is triply regular (TR). All non-trivial non-symmetric designs with λ = 1, and all quasi-symmetric designs with λ = 2 are TR. The design of points and hyperplanes of the affine geometry AG (n, q), n ≥ 3, is NTR and AG(2,q) is TR. We show that the design of points and hyperplanes of AG (3, q) is characterized by its parameters as an NTR design. Also, several `local parametersʹ of an NTR design are derived, for blocks α, β with |α ∩ β| = μ, we compute the numbers of blocks γ ≠ α,β such that |α ∩ β ∩ γ| = vI (Ci), or |α ∩ γ| = |∩γ| = |β (C01), or α n yl = , Iβ n yl = 0 (Co2) and show that these parameters are independent of α, β.