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Walsh spectrum and nega spectrum of complementary arrays

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Abstract

It has been shown that all the known binary Golay complementary sequences of length \(2^m\) can be obtained by a single binary Golay complementary array of dimension m and size \(2\times 2 \times \cdots \times 2\) which can be represented by a Boolean function. However, the construction of new binary Golay complementary sequences of length \(2^m\) or Golay complementary arrays remains an open problem. In this paper, we studied the Walsh spectrum distribution and the nega spectrum distribution of the binary or quaternary Golay (Type-I) complementary array. Then, the Walsh spectrum of the binary Type-II complementary array and the nega spectrum of the binary Type-III complementary array are investigated as well. At last, the Walsh spectrum of a binary array in a complementary array set of size 4 is discussed. This work proves that binary and quaternary complementary arrays above-mentioned can only be constructed from (generalized) Boolean functions satisfying spectral values given in this paper. For instance, a binary Type-I complementary array must be bent for even m and near-bent for odd m with respect to the Walsh spectrum, and it must be negaplateaued, nega-bent or negalandscape with respect to the nega spectrum. On the other hand, constructions of new binary and quaternary complementary arrays may help us find new (generalized) Boolean functions with specific condition, such as bent or nega-bent functions.

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References

  1. Bjørstad T.E., Parker M.G., Center S.: Equivalence between certain complementary pairs of types I and III. In: Enhancing Cryptographic Primitives with Techniques from Error Correcting Codes, NATO Science for Peace and Security Series-D: Information and Communication Security, vol. 23, pp. 203–221. IOS Press (2009)

  2. Borwein P.B., Ferguson R.A.: A complete description of Golay pairs for lengths up to 100. Math. Comput. 73(246), 967–985 (2004)

  3. Carlet C., Mesnager S.: Four decades of research on bent functions. Des. Codes Cryptogr. 78(1), 5–50 (2016)

  4. Chaturvedi A., Gangopadhyay A.K.: On generalized nega-Hadamard transform. In: Quality, Reliability, Security and Robustness in Heterogeneous Networks-9th International Conference, QShine 2013, Greader Noida, India, January 11-12, 2013, Lecture Notes of the Institute for Computer Sciences, Social Informatics and Telecommunications Engineering, vol. 115, pp. 771–777. Springer (2013)

  5. Davis J.A., Jedwab J.: Peak-to-mean power control in Ofdm, Golay complementary sequences, and reed-muller codes. IEEE Trans. Inform. Theory 45(7), 2397–2417 (1999)

  6. Dickson L.E.: History of the Theory of Numbers, Volume II: Diophantine Analysis. Courier Corporation (2005)

  7. Ding J., Noshad M., Tarokh V.: Complementary lattice arrays for coded aperture imaging. J. Opt. Soc. Am. A 33(5), 863–881 (2016)

  8. Fiedler F., Jedwab J.: How do more Golay sequences arise? IEEE Trans. Inform. Theory 52(9), 4261–4266 (2006)

  9. Fiedler F., Jedwab J., Parker M.G.: A framework for the construction of Golay sequences. IEEE Trans Inform. Theory 54(7), 3114–3129 (2008)

  10. 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)

  11. Golay M.J.: Static multislit spectrometry and its application to the panoramic display of infrared spectra. J. Opt. Soc. Am 41(7), 468–472 (1951)

  12. Jedwab J., Parker M.G.: Golay complementary array pairs. Des. Codes Cryptogr. 44(1), 209–216 (2007).

    Article  MathSciNet  Google Scholar 

  13. Li C., Li N., Parker M.G.: Complementary sequence pairs of types II and III. IEICE Trans. Fundament. Electron. Commun. Comput. Sci. E95A(11), 1819–1826 (2012)

  14. Li Y., Chu W.B.: More Golay sequences. IEEE Trans. Inform. Theory 51(3), 1141–1145 (2005).

    Article  MathSciNet  Google Scholar 

  15. Medina L.A., Parker M.G., Riera C., Stănică P.: Root-hadamard transforms and complementary sequences. Cryptogr. Commun. 12(5), 1035–1049 (2020)

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

  17. Parker M.G.: Close encounters with Boolean functions of three different kinds. In: Coding theory and applications, second international castle meeting, ICMCTA 2008, Castillo de la Mota, Medina del Campo, Spain, September 15-19, 2008. Proceedings, pp. 137–153 (2008)

  18. Parker M.G.: Polynomial residue systems via unitary transforms. In: Invited talk in post-proceedings of contact forum coding theory and cryptography III, The Royal Flemish Acadamy of Belgium for Science and the Arts, Brussels (2009)

  19. Parker M.G., Riera C.: Generalised complementary arrays. In: cryptography and coding-13th IMA international conference, IMACC 2011, Oxford, December 12-15, 2011. Proceedings, Lecture Notes in Computer Science, pp. 41–60. Springer (2011)

  20. Riera C., Parker M.G.: Generalised bent criteria for Boolean functions (I). IEEE Trans. Inform. Theory 52(9), 4142–4159 (2006)

  21. Riera C., Parker M.G.: Boolean functions whose restrictions are highly nonlinear. In: 2010 IEEE information theory workshop, ITW 2010, Dublin, August 30 - September 3, 2010, pp. 1–5. IEEE (2010)

  22. Riera C., Stǎnicǎ P.: Landscape Boolean functions. Adv. Math. Commun. 13(4), 613–627 (2019)

  23. Rothaus O.S.: On bent functions. J. Combinat Theory Ser. A 20(3), 300–305 (1976)

  24. Schmidt K.U.: Quaternary constant-amplitude codes for multicode CDMA. IEEE Trans. Inform. Theory 55(4), 1824–1832 (2009)

  25. Stănică P., Gangopadhyay S., Chaturvedi A., Gangopadhyay A.K., Maitra S.: Nega-Hadamard transform, bent and negabent functions. In: sequences and their applications-SETA 2010-6th international conference, Paris, France, September 13-17, 2010. Proceedings, Lecture Notes in Computer Science, pp. 359–372. Springer (2010)

  26. Stǎnicǎ P.: A Boolean functions view on the Golay–Rudin–Shapiro sequence. J. Combinat. Number Theory 1(1), 1–12 (2019)

  27. Wang Z., Ma D., Xue E., Gong G., Budisin S.: New construction of complementary sequence (or array) sets and complete complementary codes (II) (on the second round review of IEEE transactions on information theory. arXiv:2001.04898)

  28. Wang Z., Xue E., Gong G.: New constructions of complementary sequence pairs over \(4^q\)-qam (submitted to IEEE transactions on information theory. arXiv:2003.03459)

  29. Zheng Y., Zhang X.M.: Plateaued functions. In: Advances in Cryptology-ICICS’99, Lecture Notes in Computer Science, vol. 1726, pp. 284–300 (1999)

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  1. State Key Laboratory of Integrated Service Networks, School of Cyber Engineering, Xidian University, Xi’an, China

    Jinjin Chai, Zilong Wang & Erzhong Xue

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  1. Jinjin Chai
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  2. Zilong Wang
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Correspondence to Zilong Wang.

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Communicated by J. Jedwab.

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This research is supported in part by NSFC under Grant 62172319 and U19B2021, and NSBR under Grant 2021JQ-192.

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Chai, J., Wang, Z. & Xue, E. Walsh spectrum and nega spectrum of complementary arrays. Des. Codes Cryptogr. 89, 2663–2677 (2021). https://doi.org/10.1007/s10623-021-00938-9

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