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Designs, Codes and Cryptography

Erschienen in:

14.02.2020

Six constructions of asymptotically optimal codebooks via the character sums

verfasst von: Wei Lu, Xia Wu, Xiwang Cao, Ming Chen

Erschienen in: Designs, Codes and Cryptography | Ausgabe 6/2020

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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.

Vorheriger Artikel New MDS self-dual codes over finite fields of odd characteristic

Nächster Artikel Partially APN functions with APN-like polynomial representations

Candes E., Wakin M.: An introduction to compressive sampling. IEEE Signal Process. Mag. 25(2), 21–30 (2008).CrossRef

Conway J., Harding R., Sloane N.: Packing lines, planes, etc.: packings in Grassmannian spaces. Exp. Math. 5(2), 139–159 (1996).MathSciNetMATHCrossRef

Ding C.: Complex codebooks from combinatorial designs. IEEE Trans. Inf. Theory 52(9), 4229–4235 (2006).MathSciNetMATHCrossRef

Ding C., Feng T.: A generic construction of complex codebooks meeting the Welch bound. IEEE Trans. Inf. Theory 53(11), 4245–4250 (2007).MathSciNetMATHCrossRef

Delsarte P., Goethals J., Seidel J.: Spherical codes and designs. Geom. Dedic. 67(3), 363–388 (1997).MathSciNetMATHCrossRef

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).MathSciNetMATHCrossRef

Fickus M., Mixon D., Jasper J.: Equiangular tight frames from hyperovals. IEEE Trans. Inf. Theory 62(9), 5225–5236 (2016).MathSciNetMATHCrossRef

Fickus M., Jasper J., Mixon D., Peterson J.: Tremain equiangular tight frames (2016). arXiv:1602.03490v1.

10.

Heng Z.: Nearly optimal codebooks based on generalized Jacobi sums. Discret. Appl. Math. 250, 227–240 (2018).MathSciNetMATHCrossRef

11.

Heng Z., Ding C., Yue Q.: New constructions of asymptotically optimal codebooks with multiplicative characters. IEEE Trans. Inf. Theory 63(10), 6179–6187 (2017).MathSciNetMATHCrossRef

12.

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).MathSciNetMATHCrossRef

13.

Hu H., Wu J.: New constructions of codebooks nearly meeting the Welch bound with equality. IEEE Trans. Inf. Theory 60(2), 1348–1355 (2014).MathSciNetMATHCrossRef

14.

Kovacevic J., Chebira A.: An introduction to frames. Found. Trends Signal Process. 2(1), 1–94 (2008).MATHCrossRef

15.

Li C., Qin Y., Huang Y.: Two families of nearly optimal codebooks. Des. Codes Cryptogr. 75(1), 43–57 (2015).MathSciNetMATHCrossRef

16.

Lidl R., Niederreiter H.: Finite Fields. Cambridge University Press, Cambridge (1997).MATH

17.

Luo G., Cao X.: Two constructions of asymptotically optimal codebooks via the hyper Eisenstein sum. IEEE Trans. Inf. Theory 64(10), 6498–6505 (2018).MathSciNetMATHCrossRef

18.

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).

19.

Luo G., Cao X.: Two constructions of asymptotically optimal codebooks. Crypt. Commun. (2018). https://doi.org/10.1007/s12095-018-0331-4.

20.

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).MATH

21.

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).CrossRef

22.

Rahimi F.: Covering graphs and equiangular tight frames. Ph.D. Thesis, University of Waterloo, Ontario (2016). http://hdl.handle.net/10012/10793.

23.

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

24.

Sarwate D.: Meeting the Welch Bound with Equality, pp. 63–79. Springer, New York (1999).MATH

25.

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

26.

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).CrossRef

27.

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

28.

Wu X., Lu W., Cao X., Chen M.: Two constructions of asymptotically optimal codebooks via the trace functions (2019). arXiv:1905.01815.

29.

Xia P., Zhou S., Giannakis G.: Achieving the Welch bound with difference sets. IEEE Trans. Inf. Theory 51(5), 1900–1907 (2005).MathSciNetMATHCrossRef

30.

Yu N.: A construction of codebooks associated with binary sequences. IEEE Trans. Inf. Theory 58(8), 5522–5533 (2012).MathSciNetMATHCrossRef

31.

Zhang A., Feng K.: Two classes of codebooks nearly meeting the Welch bound. IEEE Trans. Inf. Theory 58(4), 2507–2511 (2012).MathSciNetMATHCrossRef

32.

Zhang A., Feng K.: Construction of cyclotomic codebooks nearly meeting the Welch bound. Des. Codes Cryptogr. 63(2), 209–224 (2013).MathSciNetMATHCrossRef

33.

Zhou Z., Tang X.: New nearly optimal codebooks from relative difference sets. Adv. Math. Commun. 5(3), 521–527 (2011).MathSciNetMATHCrossRef

Titel: Six constructions of asymptotically optimal codebooks via the character sums
verfasst von: Wei Lu
Xia Wu
Xiwang Cao
Ming Chen
Publikationsdatum: 14.02.2020
Verlag: Springer US
Erschienen in: Designs, Codes and Cryptography / Ausgabe 6/2020
Print ISSN: 0925-1022
Elektronische ISSN: 1573-7586
DOI: https://doi.org/10.1007/s10623-020-00735-w

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