Non-symmetric nearly triply regular designs

College

College of Science

Department/Unit

Mathematics and Statistics Department

Document Type

Article

Source Title

Discrete Mathematics

Volume

151

First Page

201

Last Page

212

Publication Date

1996

Abstract

A non-symmetric 2-(v, k, 2) design D is said to be nearly triply regular (NTR) if there are positive integers /~, v~ and v 2 with v~ > v 2 such that (i) I~ c~ fll is 0 or /~, for any two distinct blocks of D (that is, D is quasi-symmetric) and (ii) let n fln 71 is 0, v t or v 2, for any three distinct blocks ct, fl, 7 of D. If these conditions hold with v~ = v2, we say that D is triply regular (TR).

All non-trivial non-symmetric designs with 2 = 1, and all quasi-symmetric designs with 2 = 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 parameters' of an NTR design are derived; for blocks ct, fl with I~n/~l=/~, we compute the numbers of blocks y~ct, fl such that [~flc~7l = v i (Ci), or let c~ y[ = [tiny[ = p (Col), or I~nvl -- p, I~nVl = 0 (Co2) and show that these parameters are independent of ct, ft.

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Digitial Object Identifier (DOI)

10.1016/0012-365X(94)00097-3

Disciplines

Mathematics

Keywords

Combinatorial designs and configurations; Graph theory

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