Finding pairwise disjoint vector pairs in F_2^n with a prescribed sequence of differences

Abstract

We consider the following question by Balister, Győri and Schelp: given $2^{n-1}$ nonzero vectors in $\ff_2^n$ with zero sum, is it always possible to partition $\ff_2^n$ into pairs such that the difference between the two elements of the $i$-th pair is equal to the $i$-th given vector? An analogous question in $\ff_p$ was resolved by Preissmann and Mischler in 2009. In this paper, we prove the conjecture in $\ff_2^n$ in the case when there are at most $n-2\log n-1$ distinct values among the given differences, and also in the case when at least a fraction $\frac{28}{29}$ of the differences are equal.