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Designs, Codes and Cryptography

Erschienen in:

28.11.2023

Jacobi sums over Galois rings of arbitrary characters and their applications in constructing asymptotically optimal codebooks

verfasst von: Deng-Ming Xu, Gang Wang, Sihem Mesnager, You Gao, Fang-Wei Fu

Erschienen in: Designs, Codes and Cryptography | Ausgabe 4/2024

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Abstract

Codebooks with small maximum cross-correlation amplitudes are used to distinguish the signals from different users in multiple access communication systems in code division. Firstly, this paper studies the Jacobi sums over Galois rings of arbitrary characteristics and completely determines their absolute values. This extends the work by Li et al. (Sci China 56(7):1457–1465, 2013), where the Jacobi sums over some Galois rings with characteristics of a prime square were discussed. It is worth mentioning that the generalization is not trivial, as the Galois rings of arbitrary characteristics have a more complicated structure. Our deterministic construction of codebooks is based on Jacobi sums over Galois rings of arbitrary characteristics, producing asymptotically optimal codebooks for the Welch bound. Finally, compared to the literature, this article proposes for the first time design of codebooks based on Jacobi sums over Galois rings. In addition, the parameters of the presented codebooks are new.

Vorheriger Artikel Jacobi polynomials and harmonic weight enumerators of the first-order Reed–Muller codes and the extended Hamming codes

Nächster Artikel LowMS: a new rank metric code-based KEM without ideal structure

Calderbank A.R., Cameron P.J., Kantor W.M., Seidel J.J.: \(\mathbb{Z} _4\)-Kerdock codes, orthogonal spreads, and extremal Euclidean line sets. Proc. Lond. Math. Soc. 75(3), 436–480 (1997).

Carlet C.: One-weight \(\mathbb{z}_4\)-linear codes. In: Proc. of an International Conference on Coding Theory Cryptography and Related Areas, Guanajuato, Mexico, pp. 57–72 (2000).

Christensen O.: An Introduction to Frames and Riesz Bases. Birkhäuser, Boston (2003).

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

Delsarte P., Goethals J.M., Seidel J.J.: Spherical codes and designs. Geometriae Dedicate 67(3), 363–388 (1997).MathSciNet

Delsarte P., Goethals J.M., Seidel J.J.: Bounds for systems of lines and Jacobi polynomials. Philips Res. Repts. 30, 91–105 (1975).

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

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

Feng T.: A new construction of perfect nonlinear functions using Galois rings. J. Comb. Des. 17(3), 229–239 (2009).MathSciNet

10.

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

11.

Fickus M., Mixon D., Tremain J.: Steiner equiangular tight frames. Linear Algebra Appl. 436(5), 1014–1027 (2012).MathSciNet

12.

Hammons A.R., Kumar P.V., Calderbank A.R., Sloane N.J.A., Solé P.: The \(\mathbb{Z} _4\)-linearity of Kerdock, Preparata, Goethals, and related codes. IEEE Trans. Inf. Theory 40(12), 301–319 (1994).

13.

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

14.

Hong S., Park H., No J., 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).MathSciNet

15.

Hou X., Leung K., Xiang Q.: New partial difference sets in \(\mathbb{Z} _{p^2}^t\) and a related problem about Galois rings. Finite Fields Appl. 7(1), 165–188 (2001).MathSciNet

16.

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

17.

Kabatyanskii G.A., Levenshtein V.I.: Bounds for packing on a sphere and in space. Probl. Inf. Transm. 14, 1–17 (1978).MathSciNet

18.

Langevin P., Solé P.: Gauss sums over quasi-Frobenius rings. In: Proc. of the Fifth International Conference on Finite Fields and Applications, pp. 329–340, (1999).

19.

Levenshtein V.I.: Bounds for packings of metric spaces and some of their applications. Probl. Cybern. 40, 43–110 (1983) (in Russian).

20.

Li S., Ge G.: Deterministic sensing matrices arising from near orthogonal systems. IEEE Trans. Inf. Theory 60(4), 2291–2302 (2014).MathSciNet

21.

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

22.

Li J., Zhu S., Feng K.: The Gauss sums and Jacobi sums over Galois ring \(GR(p^2, r)\). Sci. China 56(7), 1457–1465 (2013).MathSciNet

23.

Lidl R., Niederreiter H.: Finite Fields and Their Applications, 2nd edn Cambridge University Press, Cambridge (1997).

24.

Love D.J., Heath R.W., Strohmer T.: Grassmannian beamforming for multiple input multiple output wireless systems. IEEE Trans. Inf. Theory 49(10), 2735–2747 (2003).MathSciNet

25.

Lu W., Wu X., Cao X., Chen M.: Six constructions of asymptotically optimal codebooks via the character sums. Des. Codes Cryptogr. 88(2), 1139–1158 (2020).MathSciNet

26.

Luo G., Cao X.: Two constructions of asymptotically optimal codebooks via the hyper Eisenstein sum. IEEE Trans. Inf. Theory 64(10), 6498–6505 (2017).MathSciNet

27.

Luo G., Cao X.: New constructions of codebooks asymptotically achieving the Welch bound. In: IEEE Int. Symp. Inf. Theory, Vail, CO, USA, pp. 2346–2350 (2018).

28.

Luo G., Cao X.: Two constructions of asymptotically optimal codebooks. Cryptogr. Commun. 11(2), 825–838 (2019).MathSciNet

29.

Massey J., Mittelholzer T.: Welch\(^{,}\)s bound and sequence sets for code-division multiple-access systems. Sequences II: Methods in Communication, Security and Computer Science, pp. 63–78. Springer, New York (1999).

30.

Qian L., Cao X.: Gaussian sums, hyper Eisenstein sums, and Jacobi sums over a local ring and their applications. AAECC 34(2), 211–244 (2021).MathSciNet

31.

Qi Y., Mesnager S., Tang C.: Codebooks from generalized bent \(\mathbb{Z} _4\)-valued quadratic forms. Discret. Math. 343(3), 111736 (2020).

32.

Sarwate D.: Meeting the Welch bound with equality. In: Proc. SETA\(^{\prime }\)\(98\), pp. 79–102. Springer, Berlin (1999).

33.

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

34.

Tarokh V., Kim I.M.: Existence and construction of noncoherent unitary space-time codes. IEEE Trans. Inf. Theory 48(12), 3112–3117 (2002).MathSciNet

35.

Tian L., Li Y., Liu T., Xu C.: Constructions of codebooks asymptotically achieving the Welch bound with additive characters. IEEE Signal Process. Let. 26(4), 622–626 (2019).

36.

Wan Z.: Lectures Notes on Finite Fields and Galois Rings. Word Scientific, Singapore (2003).

37.

Wang J.: On the Jacobi sums modulo \(p^n\). J. Number Theory 39(1), 50–64 (1991).MathSciNet

38.

Wang Q., Yan Y.: Asymptotically optimal codebooks derived from generalized bent functions. IEEE Access 8, 54905–54909 (2020).

39.

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

40.

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

41.

Xiang Q., Davis J.A.: Constructions of low-rank relative differences sets in \(2\)-groups using Galois rings. Finite Fields Appl. 6(2), 130–145 (2000).MathSciNet

42.

Xu D.: Construction of mutually unbiased maximally entangled bases in \(\mathbb{C} ^{2^s}\otimes \mathbb{C} ^{2^s}\) by using Galois rings. Quantum Inf. Process. 19(6), 175 (2020).

43.

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

44.

Yu N.Y., Feng K., Zhang A.X.: A new class of near-optimal partial Fourier codebooks from an almost difference set. Des. Codes Cryptogr. 71(3), 493–501 (2014).MathSciNet

45.

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

46.

Zhang A.X., Feng K.: Construction of cyclotomic codebooks nearly meeting the Welch bound. Des. Codes Cryptogr. 63(2), 209–224 (2012).MathSciNet

47.

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

Titel: Jacobi sums over Galois rings of arbitrary characters and their applications in constructing asymptotically optimal codebooks
verfasst von: Deng-Ming Xu
Gang Wang
Sihem Mesnager
You Gao
Fang-Wei Fu
Publikationsdatum: 28.11.2023
Verlag: Springer US
Erschienen in: Designs, Codes and Cryptography / Ausgabe 4/2024
Print ISSN: 0925-1022
Elektronische ISSN: 1573-7586
DOI: https://doi.org/10.1007/s10623-023-01328-z

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Jacobi sums over Galois rings of arbitrary characters and their applications in constructing asymptotically optimal codebooks

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