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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 $d \\geq 2$. We take a combinatorial approach to this conjecture, using mutually unbiased bases (MUBs) in the following 3 constructions of equiangular lines:\n  (1) adapting a set of $d$ MUBs in $\\mathbb{C}^d$ to obtain $d^2$ equiangular lines in $\\mathbb{C}^d$,\n  (2) using a set of $d$ MUB"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1408.5169","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-08-21T22:50:04Z","cross_cats_sorted":[],"title_canon_sha256":"b01421c855ec0830f0f8db24451c319b027e8d2e45302070c2470777760ed366","abstract_canon_sha256":"4584b7541ed574463efbc0234a08369ff588ce0526c628d14ee569377d16d717"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:20:49.635229Z","signature_b64":"8GMzsDVfpSIwm7Acx+S9oW7aGcHqvCf0OmO9RnUukCSTFfABjsfj84fjTBTEDu0B1Uiv2hgi/rLX5PcK6B9ZCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cf643c896156694a10c43f2aef58850d98d804952808bd0c1796057893b1f320","last_reissued_at":"2026-05-18T02:20:49.634594Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:20:49.634594Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Constructions of complex equiangular lines from mutually unbiased bases","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Amy Wiebe, Jonathan Jedwab","submitted_at":"2014-08-21T22:50:04Z","abstract_excerpt":"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. 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