Type of the equiangular tight frames with \(n+1\) vectors in \(R^n\)
About this article
Abstract
An equiangular tight frame (ETF) is a \(d \times n\) matrix that has orthogonal rows and unit-norm columns. ETFs have applications in communications, coding theory and quantum computing. In this paper we investigate type of ETFs that have \(n+1\) vectors in \(R^n\). Also we state the connection between these frames with the complete graphs that containing \(n+1\) vertices.References
[1] A. Aldroubi, ”Portraits of frames”, Amer.Math, 123 (1995) 1661-1668. Type of equiangular tight frames with n + 1 vectors in R n 401
[2] B.G. Bodmann, V.I. Paulsen, Frames, graphs and erasures, Linear Algebra Appl 404 (2005) 118-146.
[3] P.G. Casazza, ”The art of frame theory”, Taiwanese J. Math, Vol.4, No.2, (2000), pp.129-201.
[4] Y.C. Eldar, G.D. Forney, ”Optimal tight frames and quantum measurement”, IEEE Trans. Inform, Vol.48, No.3, (2002), pp.599-610.
[5] V.K. Goyal, J. Kovacevic, J.A. Kelner, Quantized frame expansions with erasures, Appl. Comput. Harmon. Anal 10 (2001) 203-233.
View more references (11)
[6] R. Holmes, V.I. Paulsen, Optimal frames for erasures, Linear Algebra Appl 377 (2004) 31-51.
[7] M. N. Istomina and A. B. Pevnyi, ”On location of points on a sphere and the Mercedes-Benz frame [In Russian]”, Mat. Prosveschenic, Vol.3, No.11, (2007), pp.105-112.
[8] M. N. Istomina and A. B. Pevnyi, ”The Mercedes-Benz frame in the n- dimensional space [In Russian]”, Vestnik Syktyvkar. Gos. Univ, Vol.1, No.6, (2006), pp.219-222.
[9] J. Kovacevic, A. Chebira, ”Life beyond bases: the advent of frames (part i)”, IEEE Signal Process. Mag, Vol.24, No.4, (2007), pp.86-104.
[10] J. Kovacevic, A. Chebira, ”Life beyond bases: the advent of frames (part ii)”, IEEE Signal Process. Mag, Vol.24, No.5, (2007), pp.115-125.
[11] P.W.H. Lemmens, J.J. Seidel, Equiangular lines, J. Algebra 24 (1973) 494-512.
[12] J.M. Renes, Equiangular spherical codes in quantum cryptography, Quantum Inf. Comput 5 (2005) 81-92.
[13] J.M. Renes, R. Blume-Kohout, A.J. Scott, C.M. Caves, Symmetric informationally complete quantum measurements, J. Math.Phys 45 (2004) 2171-2180.
[14] T. Strohmer, R.W. Heath, ”Grassmannian frames with applications to coding and communication”, Appl. Comput. Harmon. Anal, Vol.14, No.3, (2003), pp.257-275.
[15] M.A. Sustik, J.A. Tropp, I.S. Dhillon, R.W. Heath, On the existence of equiangular tight frames, Linear Algebra Appl 426 (2007) 619-635.
[16] S.Waldron, ”On the construction of equiangular frames from graphs”, Linear Algebra Appl, Vol.431, No.11, (2009), pp.2228-2242.