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Two constructions of two-dimensional Golay complementary array sets with flexible sizes in two dimensions

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Abstract

Two-dimensional (2-D) Golay complementary array sets (GCASs) have many practical applications such as massive multiple-input multiple-output (MIMO) transmission. Despite numerous research attempts over the past years, very few works are known on the constructions of GCASs whose array size can be flexibly changed in two dimensions. In this paper, two constructions of GCASs with flexible array sizes are considered. They are both indirect constructions with the aid of existing mutually orthogonal complementary array pairs (MOCAPs) and even-shift complementary sets (ESCSs). Our key idea is to combine several arrays into one new array through horizontal and vertical concatenation. Using this proposed method, a large number of new GCASs can be obtained as long as we can select some properly known MOCAPs. In addition, the proposed constructions have better parameters compared with the existing literature.

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References

  • Avis AA, Jedwab J (2021) Three-phase Golay sequence and array triads. J Combin Theory Ser A 180:105422

    Article  MathSciNet  Google Scholar 

  • Boyd S (1986) Multitone signals with low crest factor. IEEE Trans Circ Syst 33(10):1018–1022

    Article  Google Scholar 

  • Chen HH, Yeh J-F, Suehiro N (2001) A multicarrier CDMA architecture based on orthogonal complete complementary codes for new generations of wideband wireless communications. IEEE Commun Mag 39:126–134

    Article  Google Scholar 

  • Das S, Majhi S (2020) Two-dimensional \(Z\)-complementary array code sets based on matrices of generating polynomials. IEEE Trans Signal Process 68:5519–5532

    Article  MathSciNet  Google Scholar 

  • Davis JA, Jedwab J (1999) Peak-to-mean power control in OFDM, Golay complementary sequences, and Reed–Muller codes. IEEE Trans Inf Theory 45(7):2397–2417

    Article  MathSciNet  Google Scholar 

  • Du C, Jiang Y (2024) Polyphase Golay complementary arrays: the possible sizes and new constructions. IEEE Trans Inf Theory 70(2):1422–1435

    Article  MathSciNet  Google Scholar 

  • Dymond M (1992) Barker arrays. Existence, generalization and alternatives. Ph.D. thesis, University of London

  • Farkas P, Turcsany M (2003) Two-dimensional orthogonal complete complementary codes. In: Proceedings of joint 1st workshop mobile future symposium trends communication (SympoTIC), pp 1–5

  • Fiedler F, Jedwab J, Parker MG (2008) A multi-dimensional approach for the construction and enumeration of Golay complementary sequences. J Combin Theory (Ser A) 115:753–776

    Article  MathSciNet  Google Scholar 

  • Golay M (1961) Complementary series. IRE Trans Inf Theory 7(2):82–87

    Article  MathSciNet  Google Scholar 

  • Golomb S, Taylor H (1982) Two-dimensional synchronization patterns for minimum ambiguity. IEEE Trans Inf Theory 28(4):600–604

    Article  MathSciNet  Google Scholar 

  • Groenewald JM, Maharaj BT (2007) MIMO channel synchronization using Golay complementary pairs. In: Proceedings of AFRICON, pp 1–5

  • Hershey JE, Yarlagadda R (1983) Two-dimensional synchronisation. Electron Lett 19(19):801–803

    Article  Google Scholar 

  • Jiang Y, Li F, Wang X, Li J (2019) Autocorrelation complementary matrices. In: Proceedings of 53rd Asilomar conference on signals, systems, and computers, Pacific Grove, CA, USA, pp 1596–1600

  • Lapin I, Farkas P (2016) Peak-to-average power ratio of two-dimensional orthogonal complete complementary codes. In: 2016 international symposium ELMAR, Zadar, Croatia, pp 253–256

  • Li F, Jiang Y, Du C, Wang X (2021) Construction of Golay complementary matrices and its applications to MIMO omnidirectional transmission. IEEE Trans Signal Process 69:2100–2113

    Article  MathSciNet  Google Scholar 

  • Liu T, Men X, Li Y, Chen X (2022) Constructions of 2-D Golay complementary array sets for MIMO omnidirectional transmission. IEEE Commun Lett 20:1459–1463

    Article  Google Scholar 

  • Lu AA, Gao X, Meng X, Xia XG (2020) Omnidirectional precoding for 3D massive MIMO with uniform planar arrays. IEEE Trans Wirel Commun 19(4):2628–2642

    Article  Google Scholar 

  • Lüke H (1985) Sets of one and higher dimensional Welti codes and complementary codes. IEEE Trans Aerosp Electron Syst 21(2):170–179

    Article  MathSciNet  Google Scholar 

  • Matsufuji S, Shigemitsu R, Tanada Y, Kuroyanagi N (2004) Construction of complementary arrays. In: Joint IST workshop on mobile future and symposium on trends in communication Bratislava, Slovakia, pp 78–81

  • Meng X, Gao XG, Xia X-G (2016) Omnidirectional precoding based transmission in massive MIMO systems. IEEE Trans Commun 64(1):174–186

    Article  Google Scholar 

  • Pai C, Chen C (2022) Two-dimensional Golay complementary array pairs/sets with bounded row and column sequence PAPRs. IEEE Trans Commun 70(6):3695–3707

  • Pai C, Chen C (2020) Constructions of two-dimensional Golay complementary array pairs based on generalized Boolean functions. In: IEEE international symposium on information theory, Los Angeles, pp 2931–2935

  • Pai C, Liu Z, Zhao Y, Huang Z, Chen C (2022) Designing two-dimensional complete complementary codes for omnidirectional transmission in massive MIMO Systems. In: IEEE international symposium on information theory proceedings, pp 2285–2290

  • Paterson KG (2000) Generalized Reed–Muller codes and power control in OFDM modulation. IEEE Trans Inf Theory 46(1):104–120

    Article  MathSciNet  Google Scholar 

  • Shen B, Yang Y, Ren R (2024) Three constructions of Golay complementary array sets. Adv Math Commun 18(3):696–710

    Article  MathSciNet  Google Scholar 

  • Spasojevic P, Georghiades CN (2001) Complementary sequences for ISI channel estimation. IEEE Trans Inf Theory 47(3):1145–1152

    Article  MathSciNet  Google Scholar 

  • Tseng CC, Liu C (1972) Complementary sets of sequences. IEEE Trans Inf Theory 18(5):644–652

    Article  MathSciNet  Google Scholar 

  • Turcsany M, Farkas P (2004) New 2D-MC-DS-SS-CDMA techniques based on two-dimensional orthogonal complete complementary codes. Multi-carrier spread-spectrum. Springer, Dordrecht, pp 49–56

  • Turcsany M, Farkas P, Duda P, Kralovic J (2006) Performance evaluation of two-dimensional quasi orthogonal complete complementary codes in fading channels. In: Proceedings of joint IST workshop mobile future, symposium on trends in communications (SympoTIC), pp 84–87, Jun. 2006

  • van Nee RDJ (1996) OFDM codes for peak-to-average power reduction and error correction. In: Proceedings of IEEE global telecommunication conference, London, UK, pp 740–744

  • Wang Z, Gong G (2023) Constructions of complementary sequence sets and complete complementary codes by 2-level autocorrelation sequences and permutation polynomials. IEEE Trans Inf Theory 69(7):4723–4739

    Article  MathSciNet  Google Scholar 

  • Wang Z, Ma D, Gong G, Xue E (2021) New construction of complementary sequence (or array) sets and complete complementary codes. IEEE Trans Inf Theory 67(7):4902–4928

    Article  MathSciNet  Google Scholar 

  • Weathers G, Holliday EM (1983) Group-complementary array coding for radar clutter rejection. IEEE Trans Aerosp Electron Syst 19(3):369–379

    Article  Google Scholar 

  • Zeng F, Zhang Z (2010) Two dimensional periodic complementary array sets. In: Proceedings of international conference on computational intelligence software engineering, pp 1-4

  • Zhao Y, Pai C, Huang Z, Liu Z, Chen C (2024) Two-dimensional Golay complementary array sets with arbitrary lengths for omnidirectional MIMO transmission. IEEE Trans Commun 72(2):1135–1145

    Article  Google Scholar 

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Acknowledgements

The support of the National Natural Science Foundation of China (through grant no. 12171241) and Natural Science Foundation of Jiangsu Province (through grant no. BK20230867) are greatly acknowledged.

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  1. Hongyang Xiao and Xiwang Cao contributed equally to this work.

Authors and Affiliations

  1. School of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 211106, Jiangsu, China

    Hongyang Xiao & Xiwang Cao

  2. Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles (NUAA), MIIT, Nanjing, 211106, Jiangsu, China

    Xiwang Cao

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  1. Hongyang Xiao
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  2. Xiwang Cao
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Correspondence to Xiwang Cao.

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Xiao, H., Cao, X. Two constructions of two-dimensional Golay complementary array sets with flexible sizes in two dimensions. Comp. Appl. Math. 43, 174 (2024). https://doi.org/10.1007/s40314-024-02694-7

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