Six constructions of asymptotically optimal codebooks via the character sums. (English) Zbl 1442.94001
Codebooks with small maximum cross correlation magnitude are desired in communications. In this paper, six construction of nearly optimal codebooks are devied. The first one is known in [“Constructions of codebooks asymptotically achieving the Welch bound with additive characters”, IEEE Signal Process. Lett. 26, No. 4, 622–626 (2019)]. This construction is based on some additive characters and a special set \(D_1\). The main contribution of this paper is another five construction of codebooks. The authors use multiplicative and additive characters and special sets \(D_2,D_3,D_4,D_5\) over finite fields to modify the first construction. Consequently, they obtain five new classes of asymptotically optimal codebooks with a few maximum cross correlation magnitudes. To this end, the values of some character sums are discussed in this paper.
Reviewer: Ziling Heng (Nanjing)
References:
| [1] | Candes, E.; Wakin, M., An introduction to compressive sampling, IEEE Signal Process. Mag., 25, 2, 21-30 (2008) |
| [2] | Conway, J.; Harding, R.; Sloane, N., Packing lines, planes, etc.: packings in Grassmannian spaces, Exp. Math., 5, 2, 139-159 (1996) · Zbl 0864.51012 |
| [3] | Ding, C., Complex codebooks from combinatorial designs, IEEE Trans. Inf. Theory, 52, 9, 4229-4235 (2006) · Zbl 1237.94001 |
| [4] | Ding, C.; Feng, T., A generic construction of complex codebooks meeting the Welch bound, IEEE Trans. Inf. Theory, 53, 11, 4245-4250 (2007) · Zbl 1237.94002 |
| [5] | Delsarte, P.; Goethals, J.; Seidel, J., Spherical codes and designs, Geom. Dedic., 67, 3, 363-388 (1997) · Zbl 0376.05015 |
| [6] | Fickus M., Mixon D.: Tables of the existence of equiangular tight frames (2016). arXiv:1504.00253v2. · Zbl 1359.94318 |
| [7] | Fickus, M.; Mixon, D.; Tremain, J., Steiner equiangular tight frames, Linear Algebr. Appl., 436, 5, 1014-1027 (2012) · Zbl 1252.42032 |
| [8] | Fickus, M.; Mixon, D.; Jasper, J., Equiangular tight frames from hyperovals, IEEE Trans. Inf. Theory, 62, 9, 5225-5236 (2016) · Zbl 1359.94318 |
| [9] | Fickus M., Jasper J., Mixon D., Peterson J.: Tremain equiangular tight frames (2016). arXiv:1602.03490v1. · Zbl 1369.05217 |
| [10] | Heng, Z., Nearly optimal codebooks based on generalized Jacobi sums, Discret. Appl. Math., 250, 227-240 (2018) · Zbl 1457.94234 |
| [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) · Zbl 1390.94877 |
| [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) · Zbl 1360.94375 |
| [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) · Zbl 1364.94694 |
| [14] | Kovacevic, J.; Chebira, A., An introduction to frames, Found. Trends Signal Process., 2, 1, 1-94 (2008) · Zbl 1155.94001 |
| [15] | Li, C.; Qin, Y.; Huang, Y., Two families of nearly optimal codebooks, Des. Codes Cryptogr., 75, 1, 43-57 (2015) · Zbl 1393.94321 |
| [16] | Lidl, R.; Niederreiter, H., Finite Fields (1997), Cambridge: Cambridge University Press, Cambridge |
| [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) · Zbl 1401.94250 |
| [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). doi:10.1007/s12095-018-0331-4. · Zbl 1401.94250 |
| [20] | Massey, J.; Mittelholzer, T., Welch’s Bound and Sequence Sets for Code-Division Multiple-Access Systems: Sequences II, 63-78 (1999), New York: Springer, New York · Zbl 0838.94004 |
| [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) |
| [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) · Zbl 1071.81015 |
| [24] | Sarwate, D., Meeting the Welch Bound with Equality, 63-79 (1999), New York: Springer, New York · Zbl 1013.94510 |
| [25] | Strohmer, T.; Heath, R., Grassmannian frames with applications to coding and communication, Appl. Comput. Harmon. Anal., 14, 3, 257-275 (2003) · Zbl 1028.42020 |
| [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) |
| [27] | Welch, L., Lower bounds on the maximum cross correlation of signals, IEEE Trans. Inf. Theory, 20, 3, 397-399 (1974) · Zbl 0298.94006 |
| [28] | Wu X., Lu W., Cao X., Chen M.: Two constructions of asymptotically optimal codebooks via the trace functions (2019). arXiv:1905.01815. · Zbl 1469.94174 |
| [29] | Xia, P.; Zhou, S.; Giannakis, G., Achieving the Welch bound with difference sets, IEEE Trans. Inf. Theory, 51, 5, 1900-1907 (2005) · Zbl 1237.94007 |
| [30] | Yu, N., A construction of codebooks associated with binary sequences, IEEE Trans. Inf. Theory, 58, 8, 5522-5533 (2012) · Zbl 1364.94638 |
| [31] | Zhang, A.; Feng, K., Two classes of codebooks nearly meeting the Welch bound, IEEE Trans. Inf. Theory, 58, 4, 2507-2511 (2012) · Zbl 1365.94654 |
| [32] | Zhang, A.; Feng, K., Construction of cyclotomic codebooks nearly meeting the Welch bound, Des. Codes Cryptogr., 63, 2, 209-224 (2013) · Zbl 1236.94014 |
| [33] | Zhou, Z.; Tang, X., New nearly optimal codebooks from relative difference sets, Adv. Math. Commun., 5, 3, 521-527 (2011) · Zbl 1231.94004 |
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