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Designs, Codes and Cryptography

Erschienen in:

01.07.2016

Constructions of complex equiangular lines from mutually unbiased bases

verfasst von: Jonathan Jedwab, Amy Wiebe

Erschienen in: Designs, Codes and Cryptography | Ausgabe 1/2016

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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:

(1)

adapting a set of \(d\) MUBs in \(\mathbb {C}^d\) to obtain \(d^2\) equiangular lines in \(\mathbb {C}^d\),

(2)

using a set of \(d\) MUBs in \(\mathbb {C}^d\) to build \((2d)^2\) equiangular lines in \(\mathbb {C}^{2d}\),

(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.

Vorheriger Artikel On small gaps between the elements of multiplicative subgroups of finite fields

Nächster Artikel Perfect codes in Doob graphs

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Titel: Constructions of complex equiangular lines from mutually unbiased bases
verfasst von: Jonathan Jedwab
Amy Wiebe
Publikationsdatum: 01.07.2016
Verlag: Springer US
Erschienen in: Designs, Codes and Cryptography / Ausgabe 1/2016
Print ISSN: 0925-1022
Elektronische ISSN: 1573-7586
DOI: https://doi.org/10.1007/s10623-015-0064-8

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