Weak designs were defined by Raz, Reingold and Vadhan (1999) and are
used in constructions of extractors. Roughly speaking, a weak design
is a collection of subsets satisfying some near-disjointness
properties. Constructions of weak designs with certain parameters are
given in [RRV99]. These constructions are explicit in the sense that
they require time and space polynomial in the number of subsets.
However, the constructions require time and space polynomial in the
number of subsets even when needed to output only one specific subset
out of the collection. Hence,the constructions are not explicit in a
stronger sense.

In this work we provide constructions of weak designs (with parameters
similar to the known ones) that can be carried out in space logarithmic
in the number of subsets. Moreover, our constructions are explicit even
in a stronger sense; given an index to a subset, we output the
specified subset in time and space polynomial in the size of the index.
Using our constructions, we obtain extractors similar to the RRV
extractors in terms of parameters and that can be evaluated in
logarithmic space.

Our main construction is algebraic. In order to prove the properties
of weak designs we prove some combinatorial-algebraic lemmas that may
be interesting in their own right. These lemmas regard the number of
roots of polynomials over finite fields. In particular, we prove that
the number of polynomials (over any finite field) with $k$ roots,
vanishes exponentially in $k$. In other words, we prove that the
number of roots of a random polynomial is not only bounded by its
degree (a well known fact), but furthermore, it is concentrated
exponentially around its expectation (which is 1).