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A complementary construction using mutually unbiased bases

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Abstract

We propose a construction for complementary sets of arrays that exploits a set of mutually-unbiased bases (a MUB). In particular we present, in detail, the construction for complementary pairs that is seeded by a MUB of dimension 2, where we enumerate the arrays and the corresponding set of complementary sequences obtained from the arrays by projection. We also sketch an algorithm to uniquely generate these sequences. The pairwise squared inner-product of members of the sequence set is shown to be \(\frac {1}{2}\). Moreover, a subset of the set can be viewed as a codebook that asymptotically achieves \(\sqrt {\frac {3}{2}}\) times the Welch bound.

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Notes

  1. Up to trivial symmetries and, for a fixed δ, there may be more than one choice of ℳ δ .

  2. Comparing \(F_{k}(z)F_{k}^{*}\left (z^{-1}\right )\) with F † z − 1 F(z), we see that, whilst z − 1 is on the right for the former it is on the left for the latter. This is simply because F is a S × 1 vector - there is no deeper meaning.

  3. More accurately, we should write f k (π(x)) to indicate one of 6 possible permutations but, to reduce notation, we make such a mapping implicit in this paper.

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Acknowledgments

We wish to thank the reviewers for their helpful comments on the paper.

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Authors and Affiliations

  1. State Key Laboratory of Integrated Service Networks, Xidian University, Xi’an, 710071, China

    Gaofei Wu

  2. Department of Informatics, University of Bergen, Bergen, Norway

    Matthew Geoffrey Parker

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  1. Gaofei Wu
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  2. Matthew Geoffrey Parker
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Correspondence to Matthew Geoffrey Parker.

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Wu, G., Parker, M.G. A complementary construction using mutually unbiased bases. Cryptogr. Commun. 6, 3–25 (2014). https://doi.org/10.1007/s12095-013-0095-9

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