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 ¿ Fⁿ there is a word x ¿ 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. An application of linear balancing sets is presented for designing efficient error-correcting coding schemes in which the codewords are balanced.