(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.
