On linear balancing sets

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

Abstract: Let n be an even positive integer and F be the field GF(2). A word in F <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> is called balanced if its Hamming weight is n/2. A subset C ¿ F <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> is called a balancing set if for every word y ¿ F <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> there is a word x ¿ C such that y + x is balanced. It is shown that most linear subspaces of F <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> of dimension slightly larger than 3/2 log <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> 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.