Iskanje popolnega prirejanja v F₂ⁿ z določenimi razlikami

We consider the following question by Balister, Győri and Schelp: given 2^n-1 nonzero vectors in F₂ⁿ with zero sum, is it always possible to partition the elements of F₂ⁿ into pairs such that the difference between the two elements of the i-th pair is equal to the i-th given vector for every i? An analogous question in Fₚ, which is a case of the so-called "seating couples" problem, has been resolved by Preissmann and Mischler in 2009. In this paper, we prove the conjecture in F₂ⁿ in the case when the number of distinct values among the given difference vectors is at most n-2log (n) -1, and also in the case when at least a fraction 1/2+ε of the given vectors are equal (for all ε>0 and n sufficiently large based on ε).