The minimum volume of subspace trades

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A subspace bitrade of type Tq(t,k,v) is a pair (T₀,T₁) 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 T₀ and T₁. In a previous paper, the minimum cardinality of a subspace Tq(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 Tq(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.