Search problems in vector spaces

Abstract

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 ndimensional 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 U and NO if v 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.

Higgledy-piggledy subspaces and uniform subspace designs

“…Héger, Patkós and Takáts [1] had the idea to search generator set with respect to hyperplanes as the union of some disjoint projective lines. The generalization of this idea is to search generator set with respect to co-k-subspaces as the union of some (possibly disjoint) projective k-spaces.…”

“…The terminology 'higgledy-piggledy arrangement' is introduced by Héger, Patkós and Takáts [1] in the case of 'higgledy-piggledy lines'.…”

“…The subspace H(t) is co-ordinatized by the Plücker coordinate vector H(t) = a [0] (t) ∧ a [1] (t) ∧ · · · ∧ a [k] (t), where the i-th coordinate of the vector a [n] (t) is a i (t) = h(i, n) · t i−n , where h(i, n) ∈ F is independent from t and ∀i : h(i, 0) = 1, thus a [0] (t) = a(t). (If h(i, n) = 0 for some i < n, then the case t = 0 shall be handled separately.…”

“…Consider the collection {H(t) | t ∈ F} of subspaces co-ordinatized by the Plücker coordinate vectors H(t) = a [0] (t) ∧ a [1] (t) ∧ · · · ∧ a [k] (t). Then the orthogonal complementary subspace of H(t) is the intersection of hyperplanes:…”

“…In our previous article [4], we examined sets G of points such that each hyperplane Π is spanned by the intersection Π∩G. Examination this question had been inspired by Héger, Patkós and Takáts [1], who hunt for a set G of points in the projective space PG(d, q) that 'determines' all hyperplanes in the sense that the intersection Π ∩ G is individual for each hyperplane Π.…”