Equiangular Tight Frames From Hyperovals
An equiangular tight frame (ETF) is a set of equal norm vectors in a Euclidean space whose coherence is as small as possible, equaling the Welch bound. Also known as Welch-bound-equality sequences, such frames arise in various applications, such as waveform design, quantum information theory, compre...
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| Veröffentlicht in: | IEEE transactions on information theory 2016-09, Vol.62 (9), p.5225-5236 |
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| creator | Fickus, Matthew Mixon, Dustin G. Jasper, John |
| description | An equiangular tight frame (ETF) is a set of equal norm vectors in a Euclidean space whose coherence is as small as possible, equaling the Welch bound. Also known as Welch-bound-equality sequences, such frames arise in various applications, such as waveform design, quantum information theory, compressed sensing, and algebraic coding theory. ETFs seem to be rare, and only a few methods of constructing them are known. In this paper, we present a new infinite family of complex ETFs that arises from hyperovals in finite projective planes. In particular, we give the first ever construction of a complex ETF of 76 vectors in a space of dimension 19. Recently, a computer-assisted approach was used to show that real ETFs of this size do not exist, resolving a longstanding open problem in this field. Our construction is a modification of a previously known technique for constructing ETFs from balanced incomplete block designs. |
| doi_str_mv | 10.1109/TIT.2016.2587865 |
| format | Article |
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Also known as Welch-bound-equality sequences, such frames arise in various applications, such as waveform design, quantum information theory, compressed sensing, and algebraic coding theory. ETFs seem to be rare, and only a few methods of constructing them are known. In this paper, we present a new infinite family of complex ETFs that arises from hyperovals in finite projective planes. In particular, we give the first ever construction of a complex ETF of 76 vectors in a space of dimension 19. Recently, a computer-assisted approach was used to show that real ETFs of this size do not exist, resolving a longstanding open problem in this field. 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Our construction is a modification of a previously known technique for constructing ETFs from balanced incomplete block designs.</description><subject>Algebra</subject><subject>Coding theory</subject><subject>Coherence</subject><subject>Compressed sensing</subject><subject>Construction</subject><subject>Equiangular tight frame</subject><subject>Error correction</subject><subject>Error correction codes</subject><subject>Euclidean geometry</subject><subject>Euclidean space</subject><subject>Frames</subject><subject>Hilbert space</subject><subject>Information theory</subject><subject>Mathematical analysis</subject><subject>Norms</subject><subject>Quantum mechanics</subject><subject>Quantum theory</subject><subject>Vectors (mathematics)</subject><subject>Waveforms</subject><subject>Welch bound</subject><issn>0018-9448</issn><issn>1557-9654</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2016</creationdate><recordtype>article</recordtype><sourceid>RIE</sourceid><recordid>eNpdkE1rwkAQhpfSQlPbe6EXoRcvsTv7nWMRrYLQS3pe1mRiI4nRXVPw33cl0kNPLwPPO8w8hDwDnQLQ7C1f5VNGQU2ZNNooeUMSkFKnmZLiliSUgkkzIcw9eQhhF0chgSVkMj_2tdtv-8b5cV5vv0_jhXcthhhdO16eD-i7H9eER3JXxcCna47I12Kez5bp-vNjNXtfpwVn4pQyzSUoAxpVwUAXpnIMSpptwDFdKY20zDSTleIgqoIyDhvOuXMlYrxHCj4ik2HvwXfHHsPJtnUosGncHrs-WDBcKsqpMhF9_Yfuut7v43WRAi1lfFJFig5U4bsQPFb24OvW-bMFai_qbFRnL-rsVV2svAyVGhH_cC2pEUbzXyFeZwY</recordid><startdate>201609</startdate><enddate>201609</enddate><creator>Fickus, Matthew</creator><creator>Mixon, Dustin G.</creator><creator>Jasper, John</creator><general>IEEE</general><general>The Institute of Electrical and Electronics Engineers, Inc. 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| subjects | Algebra Coding theory Coherence Compressed sensing Construction Equiangular tight frame Error correction Error correction codes Euclidean geometry Euclidean space Frames Hilbert space Information theory Mathematical analysis Norms Quantum mechanics Quantum theory Vectors (mathematics) Waveforms Welch bound |
| title | Equiangular Tight Frames From Hyperovals |
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