The minimum volume of subspace trades

A subspace bitrade of type T-q(t, k, v) is a pair (T-0, T-1) of two disjoint nonempty collections of k-dimensional subspaces of a v-dimensional space V over the finite field of order q such that every t-dimensional subspace of V is covered by the same number of subspaces from T-0 and T-1. In a previous paper, the minimum cardinality of a subspace T-q(t, t + 1, v) bitrade was established. We generalize that result by showing that for admissible v, t, and k, the minimum cardinality of a subspace T-q(t, k, v) bitrade does not depend on k. An example of a minimum bitrade is represented using generator matrices in the reduced echelon form. For t = 1, the uniqueness of a minimum bitrade is proved. (C) 2017 Elsevier B.V. All rights reserved.