Non-symmetric nearly triply regular designs

A non-symmetric 2-(v,k,lambda) design D is said to be nearly triply regular (NTR) if there are positive integers mu, v(1) and v(2) with v(1) > v(2) such that (i) \alpha boolean AND beta\ is 0 or mu, for any two distinct blocks of D (that is, D is quasi-symmetric) and (ii) \alpha boolean AND beta boolean AND gamma\ is 0, v(1) or v(2), for any three distinct blocks alpha, beta, gamma of D. If these conditions hold with v(1) = v(2), we say that D is triply regular (TR).All non-trivial non-symmetric designs with lambda = 1, and all quasi-symmetric designs with lambda = 2 are TR. The design of points and hyperplanes of the affine geometry AG(n, q), n greater than or equal to 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 parameters' of an NTR design are derived; for blocks alpha,beta with \alpha boolean AND beta\ = mu, we compute the numbers of blocks gamma not equal alpha,beta such that \alpha boolean AND beta boolean AND gamma = v(i) (C-i), or \alpha boolean AND gamma\ = \beta boolean AND gamma\ = mu(c(01)), or \alpha boolean AND gamma\ = mu, \beta boolean AND gamma\ = 0 (C-02) and show that these parameters are independent of alpha, beta.