Search problems in vector spaces

We consider the following q-analog of the basic combinatorial search problem: let q be a prime power and GF(q) the finite field of q elements. Let V denote an n-dimensional vector space over GF(q) and let v be an unknown 1-dimensional subspace of V . We will be interested in determining the minimum number of queries that is needed to find v provided all queries are subspaces of V and the answer to a query U is YES if v 6 U and NO if v ̸6 U . This number will be denoted by A(n, q) in the adaptive case (when for each queries answers are obtained immediately and later queries might depend on previous answers) and M(n, q) in the non-adaptive case (when all queries must be made in advance). In the case n = 3 we prove 2q− 1 = A(3, q) < M(3, q) if q is large enough. While for general values of n and q we establish the bounds n log q ≤ A(n, q) ≤ (1 + o(1))nq and (1− o(1))nq ≤ M(n, q) ≤ 2nq, provided q tends to infinity. Mathematics Subject Classification (2000) 68P10 · 51E20