Root-Hadamard transforms and complementary sequences

Abstract

In this paper we define a new transform on (generalized) Boolean functions, which generalizes the Walsh-Hadamard, nega-Hadamard, 2k-Hadamard, consta-Hadamard and all HN-transforms. We describe the behavior of what we call the root-Hadamard transform for a generalized Boolean function f in terms of the binary components of f. Further, we define a notion of complementarity (in the spirit of the Golay sequences) with respect to this transform and furthermore, we describe the complementarity of a generalized Boolean set with respect to the binary components of the elements of that set.

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Acknowledgments

The authors express their deep appreciation to the editors for promptly handling our paper, as well as to the anonymous referees, whose thorough reading and constructive comments have improved the paper. The research of the first named author was supported by The Puerto Rico Science, Technology and Research Trust under agreement number 2020-00124. This content is only the responsibility of the authors and does not necesarily represent the official views of The Puerto Rico Science, Technology and Research Trust.

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Correspondence to Pantelimon Stănică.

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Dedicated to the memory of our friend and co-author, Francis N. Castro.

This article belongs to the Topical Collection: Boolean Functions and Their Applications IV

Guest Editors: Lilya Budaghyan and Tor Helleseth

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Medina, L.A., Parker, M.G., Riera, C. et al. Root-Hadamard transforms and complementary sequences. Cryptogr. Commun. (2020). https://doi.org/10.1007/s12095-020-00440-4

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Keywords

  • Golay pairs
  • Boolean functions
  • Correlations
  • Generalized root-transforms
  • Complementary sets

Mathematics Subject Classification (2010)

  • 06E30
  • 31B83
  • 94A55
  • 94C10