The minimum volume of subspace trades

TL;DR

This paper investigates the minimal size of subspace bitrades in finite vector spaces, generalizing previous results and proving uniqueness for the case when t=1, with implications for combinatorial design theory.

Contribution

It generalizes the minimum cardinality results of subspace bitrades to broader parameters and establishes the uniqueness for the case t=1.

Findings

Minimum size of T_q(t,k,v) bitrades is independent of k for admissible parameters.

Constructive example of minimum bitrade using generator matrices.

Proved uniqueness of minimum bitrade when t=1.

Abstract

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.