On linear balancing sets

Arya Mazumdar, Ron M. Roth, Pascal O. Vontobel

Abstract: Let $n$ be an even positive integer and $\mathbb F$ be the field GF$(2)$. A word in $\mathbb F$n is called balanced if its Hamming weight is $n$/$2$. A subset $\mathcal C\subseteq\mathbb F$n is called a balancing set if for every word $\mathbf y\in\mathbb F$n there is a word $\mathbf x\in \mathcal C$ such that $\mathbf y + \mathbf x$ is balanced. It is shown that most linear subspaces of $\mathbb F$n of dimension slightly larger than $\frac{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 $\mathbb F$n spans a balancing set, is NP-hard. An application of linear balancing setsis presented for designing efficient error-correcting coding schemes in which the codewords are balanced.