On the solvability of systems of bilinear equations in finite fields

Given k k sets A i ⊆ F q d A_i ⊆ F_q^d and a non-degenerate bilinear form B B in F q d F_q^d , we consider the system of l ≤ ( k 2 ) l ≤ {k}{2} bilinear equations \[ B ( a i , a j ) = λ i j , a i ∈ A i , i = 1 , … , k . B ( {a}_i, {a}_j) = λ i j, {a}_i ∈ A_i, i = 1, … , k. \] We show that the system is solvable for any λ i j ∈ F q ∗