Abstract
A set of vectors of equal norm in \(\mathbb{C}^{d}\) represents equiangular lines if the magnitudes of the 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 dāā„ā2. We describe a new construction for maximum-sized sets of equiangular lines, exposing a previously unrecognized connection with Hadamard matrices. The construction produces a maximum-sized set of equiangular lines in dimensions 2, 3 and 8.
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Acknowledgements
We thank Matt DeVos for his interest in this construction and the resulting helpful discussion and important insight regarding the proof of TheoremĀ 3.2. We are grateful to Huangjun Zhu for his generosity in pointing out the unitary transformation involvingĀ (1).
J.Ā Jedwab is supported by an NSERC Discovery Grant.
A.Ā Wiebe was supported by an NSERC Canada Graduate Scholarship.
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Dedicated to Hadi Kharaghani on the occasion on his 70th birthday
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Jedwab, J., Wiebe, A. (2015). A Simple Construction of Complex Equiangular Lines. In: Colbourn, C. (eds) Algebraic Design Theory and Hadamard Matrices. Springer Proceedings in Mathematics & Statistics, vol 133. Springer, Cham. https://doi.org/10.1007/978-3-319-17729-8_13
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DOI: https://doi.org/10.1007/978-3-319-17729-8_13
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