On linear balancing sets

TL;DR

This paper investigates the properties of linear balancing sets in binary vector spaces, showing most large subspaces are balancing sets, and explores their computational complexity and applications in error-correcting codes.

Contribution

It proves most large linear subspaces are balancing sets, generalizes to almost balancing subspaces, and establishes NP-hardness of recognizing balancing sets, with applications in coding.

Findings

Most linear subspaces of dimension > 1.5 log2(n) are balancing sets.

NP-hardness of deciding if a set spans a balancing set.

Application in designing balanced error-correcting codes.

Abstract

Let n be an even positive integer and F be the field \GF(2). A word in F^n is called balanced if its Hamming weight is n/2. A subset C \subseteq F^n$ is called a balancing set if for every word y \in F^n there is a word x \in C such that y + x is balanced. It is shown that most linear subspaces of F^n 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 F^n 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.