Density and Universality

Abstract

In many applications (testing logical circuits, construction of k-wise independent random variables, etc.), vector sets A⊆{0,1}ⁿ with the following property play an important role:

For any subset of k coordinates S={i ₁,…,i _k } the projection of A onto the indices in S contains all possible 2^k configurations.

Such sets are called (n,k)-universal. If the same holds not for all but only for at least one subset S of k indices, then A is called (n,k)-dense. The maximal number k, for which A is (n,k)-dense, is also known as the Vapnik–Chervonenkis dimension of A.