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, 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.

[1]  John I. Haas,et al.  Achieving the orthoplex bound and constructing weighted complex projective 2-designs with Singer sets , 2015, 1509.05333.

[2]  Tilen Marc,et al.  There is no (75,32,10,16) strongly regular graph , 2015, Linear Algebra and its Applications.

[3]  Hidemitsu Ogawa,et al.  Pseudoframes for subspaces with applications , 1998, Optics & Photonics.

[4]  Georgios B. Giannakis,et al.  Achieving the Welch bound with difference sets [optimal complex codebook design applications] , 2005, Proceedings. (ICASSP '05). IEEE International Conference on Acoustics, Speech, and Signal Processing, 2005..

[5]  J. J. Seidel,et al.  Geometry and Combinatorics: Selected Works of J.J. Seidel , 1991 .

[6]  Cunsheng Ding,et al.  A Generic Construction of Complex Codebooks Meeting the Welch Bound , 2007, IEEE Transactions on Information Theory.

[7]  Dursun A. Bulutoglu,et al.  E(s2)-Optimal supersaturated designs with good minimax properties , 2007 .

[8]  Thomas Strohmer,et al.  A note on equiangular tight frames , 2008 .

[9]  Boxin Tang,et al.  A method for constructing supersaturated designs and its Es2 optimality , 1997 .

[10]  R. A. Rankin,et al.  On the minimal points of positive definite quadratic forms , 1956 .

[11]  W. Wardlaw,et al.  Matrix Representation of Finite Fields , 1994 .

[12]  Wei-Hsuan Yu,et al.  There are no 76 equiangular lines in $R^{19}$ , 2015, 1511.08569.

[13]  Thomas Strohmer,et al.  GRASSMANNIAN FRAMES WITH APPLICATIONS TO CODING AND COMMUNICATION , 2003, math/0301135.

[14]  J. J. Seidel.  Geometry and Combinatorics , 2014 .

[15]  Lloyd R. Welch,et al.  Lower bounds on the maximum cross correlation of signals (Corresp.) , 1974, IEEE Trans. Inf. Theory.

[16]  Simon Lyngby Kokkendorff,et al.  Geometry and Combinatorics , 2002 .

[17]  R. Denniston.  Some maximal arcs in finite projective planes , 1969 .

[18]  H. Ogawa,et al.  Pseudoframes for Subspaces with Applications , 2004 .

[19]  Nam-Ky Nguyen.  An algorithmic approach to constructing supersaturated designs , 1996 .

[20]  V. Paulsen,et al.  Optimal frames for erasures , 2004 .

[21]  Leslie R. Welch.  Lower bounds on the maximum correlation of signals , 1974 .

[22]  Dustin G. Mixon,et al.  Kirkman Equiangular Tight Frames and Codes , 2013, IEEE Transactions on Information Theory.

[23]  Joseph M. Renes,et al.  Symmetric informationally complete quantum measurements , 2003, quant-ph/0310075.

[24]  J. J. Seidel,et al.  Strongly Regular Graphs Derived from Combinatorial Designs , 1970, Canadian Journal of Mathematics.

[25]  Joseph M. Renes.  Equiangular tight frames from Paley tournaments , 2007 .

[26]  Dustin G. Mixon,et al.  Tables of the existence of equiangular tight frames , 2015, ArXiv.

[27]  Joseph M. Renes.  Equiangular Tight Frames from Paley Tournaments , 2004 .

[28]  Mátyás A. Sustik,et al.  On the existence of equiangular tight frames , 2007 .

[29]  Dustin G. Mixon,et al.  The Road to Deterministic Matrices with the Restricted Isometry Property , 2012, Journal of Fourier Analysis and Applications.

[30]  Georgios B. Giannakis,et al.  Achieving the Welch bound with difference sets , 2005, IEEE Transactions on Information Theory.

[31]  Shayne Waldron,et al.  On the construction of equiangular frames from graphs , 2009 .

[32]  Andries E. Brouwer,et al.  Strongly Regular Graphs , 2022 .

[33]  G. Mullen,et al.  Primitive polynomials over finite fields , 1992 .

[34]  G. Zauner,et al.  QUANTUM DESIGNS: FOUNDATIONS OF A NONCOMMUTATIVE DESIGN THEORY , 2011 .

[35]  Dustin G. Mixon,et al.  Tremain equiangular tight frames , 2016, J. Comb. Theory, Ser. A.

[36]  Dustin G. Mixon,et al.  Two are better than one: Fundamental parameters of frame coherence , 2011, 1103.0435.

[37]  R. Turyn.  Character sums and difference sets. , 1965 .

[38]  Dustin G. Mixon,et al.  Steiner equiangular tight frames , 2010, 1009.5730.