On linear balancing sets

Let n be an even positive integer and F be the field GF(2). A word in Fⁿ is called balanced if its Hamming weight is n/2. A subset C ⊆ Fⁿ is called a balancing set if for every word y ε 2 Fⁿ there is a word x ε C such that y + x is balanced. It is shown that most linear subspaces of Fn of dimension slightly larger than 3/2 log₂ n are balancing sets. A generalization of this result to linear subspaces that are "almost balancing" is also presented. On the other hand, it is shown that the problem of deciding whether a given set of vectors in Fⁿ spans a balancing set, is NP-hard. An application of linear balancing sets is presented for designing efficient error-correcting coding schemes in which the codewords are balanced.