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Lower bounds on the maximum cross-correlations of 2-D quasi-complementary array sets

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Abstract

For one-dimensional (1-D) sequences, many lower bounds on the maximum cross-correlations have been demonstrated. For example, bounds proposed by Welch, Levenstein, Liu et al., and others are the lower bounds on the maximum cross-correlations of aperiodic 1-D sequence sets or quasi-complementary sequence sets (QCSSs). However, in recent times, two-dimensional (2-D) arrays have emerged with promising applications in wireless communication, such as ultra wide-band (UWB), 2-D synchronization, massive multiple-input multiple-output (MIMO), 2-D multi-carrier code division multiple access (2D-MC-CDMA), etc. Although the construction of a 2-D quasi-complementary array set (QCAS) exists in literature, the lower bound on the maximum cross-correlation \(\delta _{max}\) of such a 2-D QCAS has not been reported previously. In this paper, we propose, for the first time lower bounds on the maximum cross-correlations of 2-D QCASs for both periodic and aperiodic cases. The existing lower bounds on the maximum cross-correlations of 1-D QCSSs and 1-D sequence sets can be deduced from the proposed lower bounds on the maximum cross-correlations of 2-D QCASs and 2-D array sets for certain cases.

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Acknowledgements

The authors would like to thank the financial supports for this work. The work of Abhishek Roy was supported in parts by the Senior Research Fellowship of CSIR-HRDG, Govt. of India with file number 09/1023(0025)/2018-EMR-I, and the work of Sudhan Majhi was supported by the MATRICS project under SERB, Govt. of India with file number MTR/2020/000238(Ver-1) and the EEQ project under SERB, Govt. of India with file number EEQ/2018/000201.

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  1. Department of Mathematics, Indian Institute of Technology, Bihta, Patna, 801106, Bihar, India

    Abhishek Roy

  2. Department of Electrical Communication Engineering (ECE), Indian Institute of Science (IISc), Bangalore, 560012, Karnataka, India

    Sudhan Majhi

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  1. Abhishek Roy
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Roy, A., Majhi, S. Lower bounds on the maximum cross-correlations of 2-D quasi-complementary array sets. Cryptogr. Commun. 16, 229–247 (2024). https://doi.org/10.1007/s12095-023-00665-z

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