On linear balancing sets

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

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

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 ⊆ Fn is called a balancing set if for every word y ε 2 Fn there is a word x ε 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. 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.

Original languageEnglish (US)
Pages (from-to)345-361
Number of pages17
JournalAdvances in Mathematics of Communications
Volume4
Issue number3
DOIs
StatePublished - Aug 2010

Keywords

  • Balanced codewords
  • Balancing sets
  • Covering of hamming space
  • Linear codes

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