Abstract
A set of vectors of equal norm in \(\mathbb {C}^d\) represents equiangular lines if the magnitudes of the Hermitian inner product of every pair of distinct vectors in the set are equal. The maximum size of such a set is \(d^2\), and it is conjectured that sets of this maximum size exist in \(\mathbb {C}^d\) for every \(d \ge 2\). We take a combinatorial approach to this conjecture, using mutually unbiased bases (MUBs) in the following three constructions of equiangular lines:
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(1)
adapting a set of \(d\) MUBs in \(\mathbb {C}^d\) to obtain \(d^2\) equiangular lines in \(\mathbb {C}^d\),
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(2)
using a set of \(d\) MUBs in \(\mathbb {C}^d\) to build \((2d)^2\) equiangular lines in \(\mathbb {C}^{2d}\),
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(3)
combining two copies of a set of \(d\) MUBs in \(\mathbb {C}^d\) to build \((2d)^2\) equiangular lines in \(\mathbb {C}^{2d}\).
For each construction, we give the dimensions \(d\) for which we currently know that the construction produces a maximum-sized set of equiangular lines.
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Notes
A function \(f\) from \(\mathbb {N}\) to \(\mathbb {R}^+\) is \(\Theta (d^2)\) if there are positive constants \(c\) and \(C\), independent of \(d\), for which \(c d^2 \le f(d) \le C d^2\) for all sufficiently large \(d\).
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Acknowledgments
J. Jedwab was supported by an NSERC Discovery Grant. A. Wiebe was supported by an NSERC Canada Graduate Scholarship.
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Communicated by C. J. Colbourn.
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Jedwab, J., Wiebe, A. Constructions of complex equiangular lines from mutually unbiased bases. Des. Codes Cryptogr. 80, 73–89 (2016). https://doi.org/10.1007/s10623-015-0064-8
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DOI: https://doi.org/10.1007/s10623-015-0064-8