# American Institute of Mathematical Sciences

doi: 10.3934/amc.2022019
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## Three constructions of Golay complementary array sets

 School of Mathematics, Southwest Jiaotong University, Chengdu 611756, China

* Corresponding author: Ruibin Ren, Email: Airy_Ren@163.com

Received  October 2021 Revised  February 2022 Early access March 2022

Fund Project: The work was supported in part by the National Science Foundation of China (NSFC) under Grants 62171389 and 12102369

Recently, two-dimensional (2-D) arrays with good correlation have been used in MIMO systems. In this paper, we investigate new 2-D Golay complementary array sets (GCASs), whose 2-D aperiodic auto-correlation sums are zero for all 2-D nonzero shifts. Firstly, based on the 2-D generalized Boolean functions, we propose a direct construction of new GCASs. Secondly, using horizontal concatenation, we give two indirect constructions of GCASs. The proposed constructions can provide a lot of GCASs with flexible parameters.

Citation: Bingsheng Shen, Yang Yang, Ruibin Ren. Three constructions of Golay complementary array sets. Advances in Mathematics of Communications, doi: 10.3934/amc.2022019
##### References:
 [1] M. J. E. Golay, Multislit spectrometry, J. Opt. Sot. Amer., 39 (1949), 437-444. [2] M. J. E. Golay, Complementary series, IRE Trans. Inf. Theory, 7 (1961), 82-87.  doi: 10.1109/tit.1961.1057620. [3] R. Craigen, W. Holzmann and H. Kharaghani, Complex Golay sequences: Structure and applications, Discrete Math., 252 (2002), 73-89.  doi: 10.1016/S0012-365X(01)00162-5. [4] C. C. Tseng and C. L. Liu, Complementary sets of sequences, IEEE Trans. Inf. Theory, 18 (1972), 644-652.  doi: 10.1109/tit.1972.1054860. [5] P. Spasojević and C. N. Georghiades, Complementary sequences for ISI channel estimation, IEEE Trans. Inf. Theory, 47 (2001), 1145-1152.  doi: 10.1109/18.915670. [6] J. M. Groenewald and B. T. Maharaj, MIMO channel synchronization using Golay complementary pairs, AFRICON 2007, Windhoek, (2007), 1–5. doi: 10.1109/AFRCON.2007.4401609. [7] A. Pezeshki, A. R. Calderbank, W. Moran and S. D. Howard, Doppler resilient Golay complementary waveforms, IEEE Trans. Inf. Theory, 54 (2008), 4254-4266.  doi: 10.1109/TIT.2008.928292. [8] J. Tang, N. Zhang, Z. Ma and B. Tang, Construction of doppler resilient complete complementary code in MIMO radar, IEEE Trans. Signal Process., 62 (2014), 4704-4712.  doi: 10.1109/TSP.2014.2337272. [9] J. A. Davis and J. Jedwab, Peak-to-mean power control in OFDM, Golay complementary sequences and Reed-Muller codes, IEEE Trans. Inf. Theory, 45 (1999), 2397-2417.  doi: 10.1109/18.796380. [10] K. G. Paterson, Generalized Reed-Muller codes and power control in OFDM modulation, IEEE Trans. Inf. Theory, 46 (2000), 104-120.  doi: 10.1109/18.817512. [11] Y. Zhou, Y. Yang, Z. Zhou, S. Anand, S. Hu and Y. Guan, New complementary sets with low PAPR property under spectral null constraints, IEEE Trans. Inf. Theory, 66 (2020), 7022-7032.  doi: 10.1109/TIT.2020.3024984. [12] M. Dymond, Barker Arrays: Existence, Generalization and Alternatives, Ph.D thesis, University of London, 1992. [13] S. Matsufuji, R. Shigemitsu, Y. Tanada and N. Kuroyanagi, Construction of complementary arrays, Joint IST Workshop on Mobile Future and Symp. on Trends in Commun., Bratislava, (2004), 78–81. doi: 10.1109/TIC.2004.1409503. [14] J. Jedwab and M. G. Parker, Golay complementary array pairs, Designs, Codes and Cryptography, 44 (2007), 209-216.  doi: 10.1007/s10623-007-9088-z. [15] F. Fiedler, J. Jedwab and M. G. Parker, A multi-dimensional approach for the construction and enumeration of Golay complementary sequences, J. Combinatorial Theory (Series A), 115 (2008), 753-776.  doi: 10.1016/j.jcta.2007.10.001. [16] F. Zeng and Z. Zhang, Two dimensional periodic complementary array sets, IEEE Int. Conf. on Wireless Commun., Netw. and Mobile Comput., Bratislava, Chengdu, (2010), 1–4. doi: 10.1109/WICOM.2010.5600944. [17] F. Li, Y. Jiang, C. Du and X. Wang, Construction of Golay complementary matrices and its applications to MIMO omnidirectional transmission, IEEE Trans. Signal Process., 69 (2021), 2100-2113.  doi: 10.1109/TSP.2021.3067467. [18] A. Lu, X. Gao and X. Meng, Omnidirectional precoding for 3D massive MIMO with uniform planar arrays, IEEE Trans. Wireless Commun., 19 (2020), 2628-2642.  doi: 10.1109/TWC.2020.2966973. [19] S. W. Golomb and H. Taylor, Two-dimensional synchronization patterns for minimum ambiguity, IEEE Trans. Inf. Theory, 28 (1982), 600-604.  doi: 10.1109/TIT.1982.1056526. [20] M. Turcsány and P. Farkaš, New 2D-MC-DS-SS-CDMA techniques based on two-dimensional orthogonal complete complementary codes, Multi-Carrier Spread-Spectrum, Dordrecht, (2004), 49–56. [21] P. Farkaš and M. Turcsány, Two-dimensional orthogonal complete complementary codes, Joint IST Workshop on Mobile Future and Symp. on Trends in Commun., Bratislava, (2003), 1–5. [22] C. Pai and C. Chen, Constructions of two-dimensional Golay complementary array pairs based on generalized Boolean functions, IEEE Int. Symp. Inf. Theory, Los Angeles, (2020), 2931–2935. [23] C. Pai and C. Chen, Two-dimensional Golay complementary array sets from generalized Boolean functions, preprint, arXiv: 2102.04043. [24] C.-Y. Chen, Complementary sets of non-power-of-two length for peak-to-average power ratio reduction in OFDM, IEEE Trans. Inf. Theory, 62 (2016), 7538-7545.  doi: 10.1109/TIT.2016.2613994. [25] B. Shen, Y. Yang, Y. Feng and Z. Zhou, A generalized construction of mutually orthogonal complementary sequence sets with non-power-of-two lengths, IEEE Trans. Commun., 69 (2021), 4247-4253.

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##### References:
 [1] M. J. E. Golay, Multislit spectrometry, J. Opt. Sot. Amer., 39 (1949), 437-444. [2] M. J. E. Golay, Complementary series, IRE Trans. Inf. Theory, 7 (1961), 82-87.  doi: 10.1109/tit.1961.1057620. [3] R. Craigen, W. Holzmann and H. Kharaghani, Complex Golay sequences: Structure and applications, Discrete Math., 252 (2002), 73-89.  doi: 10.1016/S0012-365X(01)00162-5. [4] C. C. Tseng and C. L. Liu, Complementary sets of sequences, IEEE Trans. Inf. Theory, 18 (1972), 644-652.  doi: 10.1109/tit.1972.1054860. [5] P. Spasojević and C. N. Georghiades, Complementary sequences for ISI channel estimation, IEEE Trans. Inf. Theory, 47 (2001), 1145-1152.  doi: 10.1109/18.915670. [6] J. M. Groenewald and B. T. Maharaj, MIMO channel synchronization using Golay complementary pairs, AFRICON 2007, Windhoek, (2007), 1–5. doi: 10.1109/AFRCON.2007.4401609. [7] A. Pezeshki, A. R. Calderbank, W. Moran and S. D. Howard, Doppler resilient Golay complementary waveforms, IEEE Trans. Inf. Theory, 54 (2008), 4254-4266.  doi: 10.1109/TIT.2008.928292. [8] J. Tang, N. Zhang, Z. Ma and B. Tang, Construction of doppler resilient complete complementary code in MIMO radar, IEEE Trans. Signal Process., 62 (2014), 4704-4712.  doi: 10.1109/TSP.2014.2337272. [9] J. A. Davis and J. Jedwab, Peak-to-mean power control in OFDM, Golay complementary sequences and Reed-Muller codes, IEEE Trans. Inf. Theory, 45 (1999), 2397-2417.  doi: 10.1109/18.796380. [10] K. G. Paterson, Generalized Reed-Muller codes and power control in OFDM modulation, IEEE Trans. Inf. Theory, 46 (2000), 104-120.  doi: 10.1109/18.817512. [11] Y. Zhou, Y. Yang, Z. Zhou, S. Anand, S. Hu and Y. Guan, New complementary sets with low PAPR property under spectral null constraints, IEEE Trans. Inf. Theory, 66 (2020), 7022-7032.  doi: 10.1109/TIT.2020.3024984. [12] M. Dymond, Barker Arrays: Existence, Generalization and Alternatives, Ph.D thesis, University of London, 1992. [13] S. Matsufuji, R. Shigemitsu, Y. Tanada and N. Kuroyanagi, Construction of complementary arrays, Joint IST Workshop on Mobile Future and Symp. on Trends in Commun., Bratislava, (2004), 78–81. doi: 10.1109/TIC.2004.1409503. [14] J. Jedwab and M. G. Parker, Golay complementary array pairs, Designs, Codes and Cryptography, 44 (2007), 209-216.  doi: 10.1007/s10623-007-9088-z. [15] F. Fiedler, J. Jedwab and M. G. Parker, A multi-dimensional approach for the construction and enumeration of Golay complementary sequences, J. Combinatorial Theory (Series A), 115 (2008), 753-776.  doi: 10.1016/j.jcta.2007.10.001. [16] F. Zeng and Z. Zhang, Two dimensional periodic complementary array sets, IEEE Int. Conf. on Wireless Commun., Netw. and Mobile Comput., Bratislava, Chengdu, (2010), 1–4. doi: 10.1109/WICOM.2010.5600944. [17] F. Li, Y. Jiang, C. Du and X. Wang, Construction of Golay complementary matrices and its applications to MIMO omnidirectional transmission, IEEE Trans. Signal Process., 69 (2021), 2100-2113.  doi: 10.1109/TSP.2021.3067467. [18] A. Lu, X. Gao and X. Meng, Omnidirectional precoding for 3D massive MIMO with uniform planar arrays, IEEE Trans. Wireless Commun., 19 (2020), 2628-2642.  doi: 10.1109/TWC.2020.2966973. [19] S. W. Golomb and H. Taylor, Two-dimensional synchronization patterns for minimum ambiguity, IEEE Trans. Inf. Theory, 28 (1982), 600-604.  doi: 10.1109/TIT.1982.1056526. [20] M. Turcsány and P. Farkaš, New 2D-MC-DS-SS-CDMA techniques based on two-dimensional orthogonal complete complementary codes, Multi-Carrier Spread-Spectrum, Dordrecht, (2004), 49–56. [21] P. Farkaš and M. Turcsány, Two-dimensional orthogonal complete complementary codes, Joint IST Workshop on Mobile Future and Symp. on Trends in Commun., Bratislava, (2003), 1–5. [22] C. Pai and C. Chen, Constructions of two-dimensional Golay complementary array pairs based on generalized Boolean functions, IEEE Int. Symp. Inf. Theory, Los Angeles, (2020), 2931–2935. [23] C. Pai and C. Chen, Two-dimensional Golay complementary array sets from generalized Boolean functions, preprint, arXiv: 2102.04043. [24] C.-Y. Chen, Complementary sets of non-power-of-two length for peak-to-average power ratio reduction in OFDM, IEEE Trans. Inf. Theory, 62 (2016), 7538-7545.  doi: 10.1109/TIT.2016.2613994. [25] B. Shen, Y. Yang, Y. Feng and Z. Zhou, A generalized construction of mutually orthogonal complementary sequence sets with non-power-of-two lengths, IEEE Trans. Commun., 69 (2021), 4247-4253.
COMPARISONS OF 2-D GCASs
 Ref. Parameters Method [17] $(N,L,M)$ 1-D $(M,N,L)$-CCCs $(4,M,N)$ two GCPs of length $M, N$, respectively $(N_1N_2,L,M)$ 1-D $(N_1,L_1)$-GCSs and 1-D $(N_2,L_2)$-GCSs [18] $(N,L_1,L_2)$ $(N,L_1)$-GCSs and GCPs of length $L_2$ [21] $(MK,M^2,K^2)$ 1-D $(M,M,M^2)$-CCCs and 1-D $(N,N,N^2)$-CCCs [23] $(2^k,2^n,2^m)$ 2-D generalized Boolean functions Theorem 1 $(4,2^n,2^{m-1}+2^v)$ 2-D generalized Boolean functions Theorem 2 $(2K,M,L_1+L_2)$ 2-D $(K,M,L_1)$-GCASs and 2-D $(K,M,L_2)$-GCASs Remark 4 $(2^k,2^n,\sum_{s=1}^{k}2^s)$ Theorem 1 and Theorem 2 Theorem 3 $(KP,M,L_1L_2)$ 2-D $(K,M,L_1)$-GCASs and $(P,L_2)$-ESCSs
 Ref. Parameters Method [17] $(N,L,M)$ 1-D $(M,N,L)$-CCCs $(4,M,N)$ two GCPs of length $M, N$, respectively $(N_1N_2,L,M)$ 1-D $(N_1,L_1)$-GCSs and 1-D $(N_2,L_2)$-GCSs [18] $(N,L_1,L_2)$ $(N,L_1)$-GCSs and GCPs of length $L_2$ [21] $(MK,M^2,K^2)$ 1-D $(M,M,M^2)$-CCCs and 1-D $(N,N,N^2)$-CCCs [23] $(2^k,2^n,2^m)$ 2-D generalized Boolean functions Theorem 1 $(4,2^n,2^{m-1}+2^v)$ 2-D generalized Boolean functions Theorem 2 $(2K,M,L_1+L_2)$ 2-D $(K,M,L_1)$-GCASs and 2-D $(K,M,L_2)$-GCASs Remark 4 $(2^k,2^n,\sum_{s=1}^{k}2^s)$ Theorem 1 and Theorem 2 Theorem 3 $(KP,M,L_1L_2)$ 2-D $(K,M,L_1)$-GCASs and $(P,L_2)$-ESCSs
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