Spectral orbits and peak-to-average power ratio of Boolean functions with respect to the {I,H,N}n transform

  • Authors:

  • Lars Eirik Danielsen;Matthew G. Parker

  • Affiliations:

  • The Selmer Center, Department of Informatics, University of Bergen, Bergen, Norway;The Selmer Center, Department of Informatics, University of Bergen, Bergen, Norway

  • Venue:
  • SETA'04 Proceedings of the Third international conference on Sequences and Their Applications
  • Year:
  • 2004

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Abstract

We enumerate the inequivalent self-dual additive codes over GF(4) of blocklength n, thereby extending the sequence A090899 in The On-Line Encyclopedia of Integer Sequences from n = 9 to n = 12. These codes have a well-known interpretation as quantum codes. They can also be represented by graphs, where a simple graph operation generates the orbits of equivalent codes. We highlight the regularity and structure of some graphs that correspond to codes with high distance. The codes can also be interpreted as quadratic Boolean functions, where inequivalence takes on a spectral meaning. In this context we define PARIHN, peak-to-average power ratio with respect to the {I,H,N}n transform set. We prove that PARIHN of a Boolean function is equivalent to the the size of the maximum independent set over the associated orbit of graphs. Finally we propose a construction technique to generate Boolean functions with low PARIHN and algebraic degree higher than 2.