Design of concatenative complete complementary codes for CCC-CDMA via specific sequences and extended Boolean functions. (English) Zbl 07923264
Summary: A complete complementary code (CCC) consists of \(M\) sequence sets with size \(M\). The sum of the auto-correlation functions of each sequence set is an impulse function, and the sum of cross-correlation functions of the different sequence sets is equal to zero. Thanks to their excellent correlation, CCCs received extensive use in engineering. In addition, they are strongly connected to orthogonal matrices. In some application scenarios, additional requirements are made for CCCs, such as recently proposed for concatenative CCC (CCCC) division multiple access (CCC-CDMA) technologies. In fact, CCCCs are a special kind of CCCs which requires that each sequence set in CCC be concatenated to form a zero-correlation-zone (ZCZ) sequence set. However, this requirement is challenging, and the literature is thin since there is only one construction in this context. We propose to go beyond the literature through this contribution to reduce the gap between their interest and our limited knowledge of CCCCs. This paper will employ novel methods for designing CCCCs and precisely derive two constructions of these objects. The first is based on perfect cross Z-complementary pair and Hadamard matrices, and the second relies on extended Boolean functions. Specifically, we highlight that optimal and asymptotic optimal CCCCs could be obtained through the proposed constructions. Besides, we shall present a comparison analysis with former structures in the literature and examples to illustrate our main results.
MSC:
| 11T06 | Polynomials over finite fields |
| 94A11 | Application of orthogonal and other special functions |
| 94A55 | Shift register sequences and sequences over finite alphabets in information and communication theory |
| 94A60 | Cryptography |
| 94D10 | Boolean functions |
Keywords:
concatenative complete complementary code; code division multiple access; cross Z-complementary pair; extended Boolean function; correlation zoneReferences:
| [1] | Fang, J. Y.; Sun, Y. H.; Wang, L.; Wang, Q., Two-weight or three-weight binary linear codes from cyclotomic mappings, Finite Fields Appl., 85, Article 102114 pp., 2023 · Zbl 1529.94045 |
| [2] | Zhao, X. B.; Li, X. P.; Wang, Q.; Yan, T. J., A family of Hermitian dual-containing constacyclic codes and related quantum codes, Quantum Inf. Process., 20, 5, 186, 2021 · Zbl 1509.94171 |
| [3] | Sun, Y. H.; Wang, Q.; Yan, T. J., The exact autocorrelation distribution and 2-adic complexity of a class of binary sequences with almost optimal autocorrelation, Cryptogr. Commun., 10, 3, 467-477, 2018 · Zbl 1387.94061 |
| [4] | Golay, M. J.E., Complementary series, IRE Trans. Inf. Theory, 7, 2, 82-87, 1961 |
| [5] | Borwein, P. B.; Ferguson, R. A., A complete description of Golay pairs for lengths up to 100, Math. Comput., 73, 246, 967-985, 2003 · Zbl 1052.11019 |
| [6] | Tseng, C. C.; Liu, C. L., Complementary sets of sequences, IEEE Trans. Inf. Theory, 18, 5, 644-652, 1972 · Zbl 0258.94004 |
| [7] | Suehiro, N.; Hatori, M., N-shift cross-orthogonal sequences, IEEE Trans. Inf. Theory, 34, 1, 143-146, 1988 |
| [8] | Eliahou, S.; Kervaire, M.; Saffari, B., A new restriction on the lengths of Golay complementary sequences, J. Comb. Theory, Ser. A, 55, 1, 49-59, 1990 · Zbl 0705.94012 |
| [9] | Craigen, R., A theory of ternary complementary pairs, J. Comb. Theory, Ser. A, 96, 2, 358-375, 2001 · Zbl 1012.94010 |
| [10] | Davis, J. A.; Jedwab, J., Peak-to-mean power control in OFDM, Golay complementary sequences and Reed-Muller codes, IEEE Trans. Inf. Theory, 45, 7, 2397-2417, 1999 · Zbl 0960.94012 |
| [11] | Fiedler, F.; Jedwab, J.; Parker, M. G., A multi-dimensional approach to the construction and enumeration of Golay complementary sequences, J. Comb. Theory, Ser. A, 115, 5, 753-776, 2008 · Zbl 1154.05012 |
| [12] | Jedwab, J.; Parker, M. G., A construction of binary Golay sequence pairs from odd-length Barker sequences, J. Comb. Des., 17, 6, 478-491, 2009 · Zbl 1233.94015 |
| [13] | Gibson, R. G.; Jedwab, J., Quaternary Golay sequence pairs II: odd length, Des. Codes Cryptogr., 59, 147-157, 2011 · Zbl 1233.94014 |
| [14] | Turyn, R., Hadamard matrices, Baumert-Hall units, four-symbol sequences, pulse compression and surface wave encodings, J. Comb. Theory, Ser. A, 16, 313-333, 1974 · Zbl 0291.05016 |
| [15] | Liu, Z. L.; Parampalli, U.; Guan, Y. L., Optimal odd-length binary Z-complementary pairs, IEEE Trans. Inf. Theory, 60, 9, 5768-5781, 2014 · Zbl 1360.94282 |
| [16] | Zhou, Y. J.; Zhou, Z. C.; Gu, Z.; Fan, P. Z., Low-PMEPR rotatable pilot sequences for MIMO-OFDM systems, Sci. China Inf. Sci., 65, 12, Article 229302 pp., 2022 |
| [17] | Tang, J.; Zhang, N.; Ma, Z. K.; Tang, B., Construction of Doppler resilient complete complementary code in MIMO radar, IEEE Trans. Signal Process., 62, 18, 4704-4712, 2014 · Zbl 1394.94587 |
| [18] | Wang, S. Q.; Abdi, A., MIMO ISI channel estimation using uncorrelated Golay complementary sets of polyphase sequences, IEEE Trans. Veh. Technol., 56, 5, 3024-3039, 2007 |
| [19] | Sun, S. Y.; Chen, H. H.; Meng, W. X., A survey on complementary-coded MIMO CDMA wireless communications, IEEE Commun. Surv. Tutor., 17, 1, 52-69, 2015 |
| [20] | Welch, L., Lower bounds on the maximum cross correlation of signals, IEEE Trans. Inf. Theory, IT-20, 3, 397-399, 1974 · Zbl 0298.94006 |
| [21] | Zhang, W. G.; Pasalic, E.; Liu, Y. R.; Zhang, L. Q.; Xie, C. L., A design and flexible assignment of orthogonal binary sequence sets for (QS)-CDMA systems, Des. Codes Cryptogr., 91, 2, 373-389, 2023 · Zbl 1520.94033 |
| [22] | Smith, D. H.; Ward, R. P.; Perkins, S., Gold codes, Hadamard partitions and the security of CDMA systems, Des. Codes Cryptogr., 51, 231-243, 2009 · Zbl 1247.05043 |
| [23] | Chen, H. H.; Yeh, J. F.; Seuhiro, N., A multi-carrier CDMA architecture based on orthogonal complementary codes for new generations of wideband wireless communications, IEEE Commun. Mag., 39, 10, 126-135, 2001 |
| [24] | Rathinakumar, A.; Chaturvedi, A. K., Complete mutually orthogonal Golay complementary sets from Reed Muller codes, IEEE Trans. Inf. Theory, 54, 3, 1339-1346, 2008 · Zbl 1279.94152 |
| [25] | Han, C. G.; Suehiro, N.; Hashimoto, T., A systematic framework for the construction of optimal complete complementary codes, IEEE Trans. Inf. Theory, 57, 9, 6033-6042, 2011 · Zbl 1365.94380 |
| [26] | Sarkar, P.; Li, C.; Majhi, S.; Liu, Z. L., Asymptotically optimal quasi-complementary code sets from multivariate functions, 2021, arXiv, Available: |
| [27] | Liu, Z. L.; Guan, Y. L.; Parampalli, U., New complete complementary codes for peak-to-mean power control in multi-carrier CDMA, IEEE Commun. Mag., 62, 3, 1105-1113, 2014 |
| [28] | Meng, W. X.; Sun, S. Y.; Chen, H. H.; Li, J. Q., Multi-user interference cancellation in complementary coded cDMA with diversity gain, IEEE Wirel. Commun. Lett., 2, 3, 303-306, 2013 |
| [29] | Mizuyoshi, H.; Han, C. G., Concatenative complete complementary code division multiple access and its fast transform, IEEE Trans. Wirel. Commun., 22, 12, 8530-8542, 2023 |
| [30] | Fan, P. Z.; Suehiro, N.; Kuroyanagi, N.; Deng, X. M., Class of binary sequences with zero correlation zone, Electron. Lett., 35, 10, 777-779, 1999 |
| [31] | Han, C. G.; Hashimoto, T., Z-connectable complete complementary codes and its application in CDMA systems, (Proc. IEEE Int. Symp. Infor. Theory (ISIT). Proc. IEEE Int. Symp. Infor. Theory (ISIT), Seoul, Korea, 2009), 438-442 |
| [32] | Tang, X. H.; Fan, P. Z.; Matsufuji, S., Lower bounds on correlation of spreading sequence set with low or zero correlation zone, Electron. Lett., 36, 6, 551-552, 2000 |
| [33] | Torii, H.; Nakamura, M.; Suehiro, N., A new class of zero-correlation zone sequences, IEEE Trans. Inf. Theory, 50, 3, 559-565, 2004 · Zbl 1288.94046 |
| [34] | Liu, Z. L.; Yang, P.; Guan, Y. L.; Xiao, P., Cross Z-complementary pairs for optimal training in spatial modulation over frequency selective channels, IEEE Trans. Signal Process., 68, 1529-1543, 2020 · Zbl 07590840 |
| [35] | Shen, B. S.; Yang, Y.; Zhou, Z. C.; Mesnager, S., Constructions of spectrally null constrained complete complementary codes via the graph of extended Boolean functions, IEEE Trans. Inf. Theory, 69, 9, 6028-6039, 2023 · Zbl 07883349 |
| [36] | Pai, C. Y.; Lin, Y. J.; Chen, C. Y., Optimal and almost-optimal Golay-ZCZ sequence sets with bounded PAPRs, IEEE Trans. Commun., 71, 2, 728-740, 2023 |
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