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Welch bound equality sets with few distinct inner products from Delsarte-Goethals sets

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Abstract

Sets of signals that meet Welch bounds with equality or near equality are of value in communications and sensing applications, and the construction of such signal sets has been an active research area. Although Welch derived a family of bounds indexed by positive integers k, only the first Welch bound (i.e., for k = 1) has been considered in these constructions. Earlier, a frame-theoretic perspective was introduced on the higher Welch bounds that is valuable in constructing signals that simultaneously meet multiple Welch bounds with equality or near equality. This perspective is used in this paper to examine the existence of signal sets that meet the kth Welch bound with equality by using second order Reed-Muller codes. Some examples of such signal sets are presented and connections to equiangular lines and t-designs are discussed.

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Notes

  1. Given a finite frame X = {x1,...,xm} for an n-dimensional complex vector space V, the function \(F:V\rightarrow \mathbb {C}^{m}\) given by \(F(w)=[\left < x_{1},w\right > \ldots \left < x_{m},w\right >]^{\mathrm {T}}\) will be called the analysis operator associated with X, while \(\mathcal {F}=F^{*} F:V\rightarrow V\) (i.e., the composition of the adjoint of F with F) will be called the frame operator associated with X.

References

  1. Welch, L.R.: Lower bounds on the maximum cross correlation of signals. IEEE Trans. Inf. Theory 20(3), 397–399 (1974)

    Article  MATH  Google Scholar 

  2. Klappenecker, A., Rötteler, M.: Mutually unbiased bases are complex projective 2-designs. In: Proceedings of the International Symposium on Information Theory, pp. 1740–1744 (2005)

  3. Strohmer, T., Heath, R.W. Jr: Grassmannian frames with applications to coding and communication. Appl. Comput. Harmon. Anal. 14(3), 257–275 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  4. Waldron, S.: Generalized Welch bound equality sequences are tight frames. IEEE Trans. Inf. Theory 49(9), 2307–2309 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. Sarwate, D.V.: Meeting the Welch Bound with Equality. DMTCS Series. Springer, Berlin (1999). Chap. 11

    MATH  Google Scholar 

  6. Xia, P., Zhou, S., Giannakis, G.B.: Achieving the Welch bound with difference sets. IEEE Trans. Inf. Theory 51(5) (2005)

  7. Ding, C., Feng, T.: A generic construction of complex codebooks meeting the Welch bound. IEEE Trans, Inf. Theory 53(11) (2007)

  8. Holmes, R.B., Paulsen, V.I.: Optimal frames for erasures. Linear Algebra Appl. 377, 31–51 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. Massey, J.L., Mittelholzer, T.: Welch’s bound and sequence sets for code-division multiple-access systems. In: Capocelli, R., De Santis, A., Vaccaro, U. (eds.) Sequences, II: Methods in Communication, Security and Computer Science (Positano, 1991), pp 63–78. Springer (1993)

  10. Datta, S., Howard, S., Cochran, D.: Geometry of the Welch bounds. Linear Algebra Appl. 437(10), 2455–2470 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. van Lint, J.H., Seidel, J.J.: Equilateral point sets in elliptic geometry. Nederl. Akad. Wetensch. Proc. Ser. A 69=Indag. Math. 28, 335–348 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  12. Delsarte, P., Goethals, J.M., Seidel, J.J.: Bounds for systems of lines and Jacobi polynomials. Philips Res. Repts. 30(3), 91–105 (1975). Issue in honour of C.J. Bouwkamp

    MATH  Google Scholar 

  13. Sustik, M.A., Tropp, J.A., Dhillon, I.S., Heath, R.W. Jr: On the existence of equiangular tight frames. Linear Algebra Appl. 426(2-3), 619–635 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Bodmann, B.G., Paulsen, V.I., Tomforde, M.: Equiangular tight frames from complex Seidel matrices containing cube roots of unity. Linear Algebra Appl. 430(1), 396–417 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Renes, J.M., Blume-Kohout, R., Scott, A.J., Caves, C.M.: Symmetric informationally complete quantum measurements. J. Math. Phys. 45(6), 2171–2180 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  16. Grassl, M.: On SIC-POVMs and MUBs in dimension 6. In: Proc. ERATO Conf. on Quant. Inf. Science (EQUIS 2004), pp 60–61 (2004)

  17. Scott, A.J., Grassl, M.: Symmetric informationally complete positive-operator-valued measures: A new computer study. J. Math. Phy. 51(4), 042203 (2010). https://doi.org/10.1063/1.3374022

    Article  MathSciNet  MATH  Google Scholar 

  18. Appleby, M., Chien, T.-Y., Flammia, S., Waldron, S.: Constructing exact symmetric informationally complete measurements from numerical solutions. J. Phys. A Math. Theor. 51(16), 165302 (2018). https://doi.org/10.1088/1751-8121/aab4cd

    Article  MathSciNet  MATH  Google Scholar 

  19. Scott, A.J.: SICs: Extending the list of solutions.arXiv:1703.03993 (2017)

  20. Hammons, A.R., Kumar, P.V., Calderbank, A.R., Sloane, N.J.A., Sole, P.: The \(\mathbb {Z}_{4}\)-linearity of Kerdock, Preparata, Goethals, and related codes. IEEE Trans. Inf. Theory 40(2), 301–319 (1994). https://doi.org/10.1109/18.312154

    Article  MathSciNet  MATH  Google Scholar 

  21. Calderbank, R., Howard, S., Jafarpour, S.: Construction of a large class of deterministic sensing matrices that satisfy a statistical isometry property. IEEE J. Sel. Top. Sig. Process. 4(2), 358–374 (2010). https://doi.org/10.1109/JSTSP.2010.2043161

    Article  Google Scholar 

  22. Kerdock, A.M.: A class of low rate nonlinear binary codes. Information and Control (1972)

  23. MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes. North-Holland Pub. Co (16) (1977)

  24. Kantor, W.M.: MUBs inequivalence and affine planes. J. Math. Phys. 53. https://doi.org/10.1063/1.3690050 (2012)

  25. Keung, R., Gross, D.: Qubit stabilizer states are complex projective 3-designs. arXiv:1510.02767 (2015)

  26. Christensen, O.: An Introduction to Frames and Riesz Bases. Applied and Numerical Harmonic Analysis, 2nd edn. Birkhäuser, Birkhäuser (2016)

    MATH  Google Scholar 

  27. Thill, M., Hassibi, B.: Group frames with few distinct inner products and low coherence. IEEE Trans. Sig. Process. 63(19), 5222–5237 (2015). https://doi.org/10.1109/TSP.2015.2450195

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The author is grateful to Doug Cochran for many useful discussions on the topic of this article. The author would like to acknowledge support from the Simons Foundation under Award Number 709212. The author also wishes to thank the anonymous referees for their suggestions.

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  1. Department of Mathematics and Statistical Science, University of Idaho, 875 Perimeter Drive, Idaho, 83843-1103, USA

    Somantika Datta

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Datta, S. Welch bound equality sets with few distinct inner products from Delsarte-Goethals sets. Cryptogr. Commun. 15, 719–729 (2023). https://doi.org/10.1007/s12095-023-00635-5

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