Vol. 103, No. 2, 1982

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ISSN: 0030-8730
On extensions of nets

Dieter Jungnickel and Sharad S. Sane

Vol. 103 (1982), No. 2, 437–455
Abstract

(s,r,μ)-nets are generalizations of the well-known Bruck nets; here any two nonparallel blocks intersect in μ points, any parallel class consists of s blocks and there are r parallel classes. We generalize the notion of transversals from the Bruck nets to the case of arbitrary μ. This notion is used to study extensions of a given net. We call a net step-t-extendable iff t new parallel classes can be adjoined. It is known that a symmetric (s,μ)-net (i.e., an (s,sμ,μ)-net whose dual is likewise an (s,sμ,μ)-net) is step-1-extendable; we show that it is step-2-extendable if and only if s divides μ and step-t-extendable (for t 3) if and only if there exists an (s,t,μ∕s)-net. We then give an alternative, matrix-free proof for the results of Shrikhande and Bhagwandas on the completion of (s,r,μ)-nets with deficiency 1 or 2. We also construct an infinite series of (4,r,μ)-nets of deficiency 2 that cannot be completed. We discuss a conjecture that would have interesting consequences for the possible parameters of affine 2-designs.

Mathematical Subject Classification 2000
Primary: 05B30
Secondary: 05B15
Milestones
Received: 31 March 1980
Revised: 13 November 1980
Published: 1 December 1982
Authors
Dieter Jungnickel
Universitat Augsburg
United States
Sharad S. Sane