On linear balancing sets (0901.3170v2)
Abstract: 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 \subseteq Fn$ is called a balancing set if for every word y \in Fn there is a word x \in C such that y + x is balanced. It is shown that most linear subspaces of Fn of dimension slightly larger than 3/2\log_2(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.