Abstract
In this paper, using additive characters of finite field, we find a codebook which is equivalent to the measurement matrix in Mohades et al. (IEEE Signal Process Lett 21(7):839–843, 2014). The advantage of our construction is that it can be generalized naturally to construct the other five classes of codebooks using additive and multiplicative characters of finite field. We determine the maximum cross-correlation amplitude of these codebooks by the properties of characters and character sums. We prove that all the codebooks we constructed are asymptotically optimal with respect to the Welch bound. The parameters of these codebooks are new.
Similar content being viewed by others
References
Candes E., Wakin M.: An introduction to compressive sampling. IEEE Signal Process. Mag. 25(2), 21–30 (2008).
Conway J., Harding R., Sloane N.: Packing lines, planes, etc.: packings in Grassmannian spaces. Exp. Math. 5(2), 139–159 (1996).
Ding C.: Complex codebooks from combinatorial designs. IEEE Trans. Inf. Theory 52(9), 4229–4235 (2006).
Ding C., Feng T.: A generic construction of complex codebooks meeting the Welch bound. IEEE Trans. Inf. Theory 53(11), 4245–4250 (2007).
Delsarte P., Goethals J., Seidel J.: Spherical codes and designs. Geom. Dedic. 67(3), 363–388 (1997).
Fickus M., Mixon D.: Tables of the existence of equiangular tight frames (2016). arXiv:1504.00253v2.
Fickus M., Mixon D., Tremain J.: Steiner equiangular tight frames. Linear Algebr. Appl. 436(5), 1014–1027 (2012).
Fickus M., Mixon D., Jasper J.: Equiangular tight frames from hyperovals. IEEE Trans. Inf. Theory 62(9), 5225–5236 (2016).
Fickus M., Jasper J., Mixon D., Peterson J.: Tremain equiangular tight frames (2016). arXiv:1602.03490v1.
Heng Z.: Nearly optimal codebooks based on generalized Jacobi sums. Discret. Appl. Math. 250, 227–240 (2018).
Heng Z., Ding C., Yue Q.: New constructions of asymptotically optimal codebooks with multiplicative characters. IEEE Trans. Inf. Theory 63(10), 6179–6187 (2017).
Hong S., Park H., Helleseth T., Kim Y.: Near optimal partial Hadamard codebook construction using binary sequences obtained from quadratic residue mapping. IEEE Trans. Inf. Theory 60(6), 3698–3705 (2014).
Hu H., Wu J.: New constructions of codebooks nearly meeting the Welch bound with equality. IEEE Trans. Inf. Theory 60(2), 1348–1355 (2014).
Kovacevic J., Chebira A.: An introduction to frames. Found. Trends Signal Process. 2(1), 1–94 (2008).
Li C., Qin Y., Huang Y.: Two families of nearly optimal codebooks. Des. Codes Cryptogr. 75(1), 43–57 (2015).
Lidl R., Niederreiter H.: Finite Fields. Cambridge University Press, Cambridge (1997).
Luo G., Cao X.: Two constructions of asymptotically optimal codebooks via the hyper Eisenstein sum. IEEE Trans. Inf. Theory 64(10), 6498–6505 (2018).
Luo G., Cao X.: New constructions of codebooks asymptotically achieving the Welch bound. In: Proceedings of IEEE International Symposium on Information Theory, Vail, CO, pp. 2346–2349 (2018).
Luo G., Cao X.: Two constructions of asymptotically optimal codebooks. Crypt. Commun. (2018). https://doi.org/10.1007/s12095-018-0331-4.
Massey J., Mittelholzer T.: Welch’s Bound and Sequence Sets for Code-Division Multiple-Access Systems: Sequences II, pp. 63–78. Springer, New York (1999).
Mohades M., Mohades A., Tadaion A.: A Reed-Solomom code based measurement matrix with small coherence. IEEE Signal Process. Lett. 21(7), 839–843 (2014).
Rahimi F.: Covering graphs and equiangular tight frames. Ph.D. Thesis, University of Waterloo, Ontario (2016). http://hdl.handle.net/10012/10793.
Renes J., Blume-Kohout R., Scot A., Caves C.: Symmetric informationally complete quantum measurements. J. Math. Phys. 45(6), 2171–2180 (2004).
Sarwate D.: Meeting the Welch Bound with Equality, pp. 63–79. Springer, New York (1999).
Strohmer T., Heath R.: Grassmannian frames with applications to coding and communication. Appl. Comput. Harmon. Anal. 14(3), 257–275 (2003).
Tian L., Li Y., Liu T., Xu C.: Constructions of codebooks asymptotically achieving the Welch bound with additive characters. IEEE Signal Process. Lett. 26(4), 622–626 (2019).
Welch L.: Lower bounds on the maximum cross correlation of signals. IEEE Trans. Inf. Theory 20(3), 397–399 (1974).
Wu X., Lu W., Cao X., Chen M.: Two constructions of asymptotically optimal codebooks via the trace functions (2019). arXiv:1905.01815.
Xia P., Zhou S., Giannakis G.: Achieving the Welch bound with difference sets. IEEE Trans. Inf. Theory 51(5), 1900–1907 (2005).
Yu N.: A construction of codebooks associated with binary sequences. IEEE Trans. Inf. Theory 58(8), 5522–5533 (2012).
Zhang A., Feng K.: Two classes of codebooks nearly meeting the Welch bound. IEEE Trans. Inf. Theory 58(4), 2507–2511 (2012).
Zhang A., Feng K.: Construction of cyclotomic codebooks nearly meeting the Welch bound. Des. Codes Cryptogr. 63(2), 209–224 (2013).
Zhou Z., Tang X.: New nearly optimal codebooks from relative difference sets. Adv. Math. Commun. 5(3), 521–527 (2011).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by T. Helleseth.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work was supported by the National Natural Science Foundation of China (Grant Nos. 11971102, 11801070, 11771007, and 61572027) and the Basic Research Foundation (Natural Science).
Rights and permissions
About this article
Cite this article
Lu, W., Wu, X., Cao, X. et al. Six constructions of asymptotically optimal codebooks via the character sums. Des. Codes Cryptogr. 88, 1139–1158 (2020). https://doi.org/10.1007/s10623-020-00735-w
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-020-00735-w