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Self-dual bent sequences for complex Hadamard matrices

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Abstract

A new notion of bent sequence related to Hadamard matrices was introduced recently, motivated by a security application (Solé et al. 2021). In this paper we introduce the analogous notion for complex Hadamard matrices, and we study the self-dual class in length at most 90. We use three competing methods of generation: Brute force, Linear Algebra and Groebner bases. Regular complex Hadamard matrices and Bush-type complex Hadamard matrices provide many examples. We introduce the strong automorphism group of complex Hadamard matrices, which acts on their associated self-dual bent sequences. We give an efficient algorithm to compute that group. We also answer the question which complex Hadamard matrices can be uniquely reconstructed from the off-diagonal elements, define a related concept of mixed-skew Hadamard matrix, and show the existence of mixed-skew Hadamard matrices of small orders.

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Notes

  1. In Table 2,“—” means that the number of all self-dual bent sequences is unknown, as the dimensions of the eigenspaces attached to the eigenvalues of C are greater than 16, we are not able to obtain all the self-dual bent sequences. In Table 2, the positive integer N in \(N^{2}\) is the number of real-valued self-dual bent sequences.

  2. https://github.com/Qomo-CHENG/Hadamard_bent_complex.

References

  1. Balonin N.A., Seberry J.: A review and new symmetric conference matrices. Informatsionno-upravlyayushchie sistemy 4(71), 2–7 (2014).

    Google Scholar 

  2. Bömer L., Antweiler M.: Periodic complementary binary sequences. IEEE Trans. Inf. Theory 36(6), 1487–1494 (1990). https://doi.org/10.1109/18.59954.

    Article  MathSciNet  MATH  Google Scholar 

  3. Bruzda W., Tadej W., Zyczkowski K.: Catalogue of Complex Hadamard Matrices. (2009). https://chaos.if.uj.edu.pl/~karol/hadamard/index.html

  4. Carlet C., Danielsen L.E., Parker M.G., Solé P.: Self-dual bent functions. Int. J. Inf. Coding Theory 1(4), 384–399 (2010). https://doi.org/10.1504/IJICoT.2010.032864.

    Article  MathSciNet  MATH  Google Scholar 

  5. Crnković D., Švob A.: Switching for 2-designs. Des. Codes Cryptogr. 90, 1585–1593 (2022). https://doi.org/10.1007/s10623-022-01059-7.

    Article  MathSciNet  MATH  Google Scholar 

  6. Frank R.: Polyphase complementary codes. IEEE Trans. Inf. Theory 26(6), 641–647 (1980). https://doi.org/10.1109/TIT.1980.1056272.

    Article  MathSciNet  Google Scholar 

  7. Golay M.J.E.: Multi-slit spectrometry. J. Opt. Soc. Am. 39(6), 437–444 (1949). https://doi.org/10.1364/JOSA.39.000437.

    Article  Google Scholar 

  8. Hammons A.R., Kumar P.V., Calderbank A.R., Sloane N.J.A., Solé P.: The \(Z_4\)-linearity of Kerdock, Preparata, Goethals, and related codes. IEEE Trans. Inf. Theory 40(2), 301–319 (1994). https://doi.org/10.1109/18.312154.

    Article  MATH  Google Scholar 

  9. Horadam K.J.: Hadamard Matrices and Their Applications. Princeton University Press, Princeton (2007).

    Book  MATH  Google Scholar 

  10. Janko Z.: The existence of a Bush-type Hadamard matrix of order 36 and two new infinite classes of symmetric designs. J. Comb. Theory Ser. A 95(2), 360–364 (2001). https://doi.org/10.1006/jcta.2000.3166.

    Article  MathSciNet  MATH  Google Scholar 

  11. Janko Z., Kharaghani H.: A block negacyclic Bush-type Hadamard matrix and two strongly regular graphs. J. Comb. Theory Ser. A 98(1), 118–126 (2002). https://doi.org/10.1006/jcta.2001.3231.

    Article  MathSciNet  MATH  Google Scholar 

  12. Kharaghani H.: On the twin designs with the ionin-type parameters. Electron. J. Comb. 7(R1), 1–11 (2000). https://doi.org/10.37236/1479.

    Article  MathSciNet  MATH  Google Scholar 

  13. Kharaghani H., Seberry J.: Regular complex Hadamard matrices. Congr. Numer. 75, 187–201 (1990).

    MathSciNet  MATH  Google Scholar 

  14. Kharaghani H., Seberry J.: The excess of complex Hadamard matrices. Graphs Comb. 9(1), 47–56 (1993). https://doi.org/10.1007/BF01195326.

    Article  MathSciNet  MATH  Google Scholar 

  15. Koukouvinos C.: Design Theory Directory. http://www.math.ntua.gr/~ckoukouv/

  16. MacWilliams F.J., Sloane N.J.A.: The Theory of Error-Correcting Codes. North Holland, Amsterdam (1977).

    MATH  Google Scholar 

  17. Moorhouse G.E.: The 2-transitive complex Hadamard matrices. http://ericmoorhouse.org/pub/complex.pdf

  18. Shi M.J., Li Y.Y., Cheng W., Crnković D., Krotov D., Solé P.: Self-dual Hadamard bent sequences. J. Syst. Sci. Complex. (2022), to appear. arXiv:2203.16439

  19. Sivaswamy R.: Multiphase complementary codes. IEEE Trans. Inf. Theory 24(5), 546–552 (1978). https://doi.org/10.1109/TIT.1978.1055936.

    Article  MATH  Google Scholar 

  20. Sok L., Shi M.J., Solé P.: Classification and construction of quaternary self-dual bent functions. Cryptogr. Commun. 10(2), 277–289 (2018). https://doi.org/10.1007/s12095-017-0216-y.

    Article  MathSciNet  MATH  Google Scholar 

  21. Solé P., Cheng W., Guilley S., Rioul O.: Bent Sequences over Hadamard Codes for Physically Unclonable Functions. IEEE International Symposium on Information Theory, 801–806 (2021) https://doi.org/10.1109/ISIT45174.2021.9517752

  22. Tadej W., Życzkowski K.: A concise guide to complex Hadamard matrices. https://arxiv.org/pdf/quant-ph/0512154.pdf

  23. Turyn R.J.: An infinite class of Williamson matrices. J. Comb. Theory Ser. A 12(3), 319–321 (1972). https://doi.org/10.1016/0097-3165(72)90095-7.

    Article  MathSciNet  MATH  Google Scholar 

  24. van Lint J.H., Wilson R.: A Course in Combinatorics, 2nd edn Cambridge University Press, Cambridge (2001).

    Book  MATH  Google Scholar 

  25. Wallis J.: Complex hadamard matrices. Linear Multilinear Algebra 1(3), 257–272 (1973). https://doi.org/10.1080/03081087308817024.

    Article  MathSciNet  MATH  Google Scholar 

  26. Zinoviev V.A., Zinoviev D.: On the generalized concatenated construction for the \(L_1\) and Lee metrics. Probl. Inf. Transm. 57(1), 70–83 (2021). https://doi.org/10.1134/S003294602101004X.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Key Laboratory of Intelligent Computing and Signal Processing of Ministry of Education, School of Mathematical Sciences, Anhui University, Hefei, 230601, Anhui, China

    Minjia Shi & Yaya Li

  2. Télécom Paris, Secure-IC S.A.S., 104 Boulevard du Montparnasse, 75014, Paris, France

    Wei Cheng

  3. Faculty of Mathematics, University of Rijeka, Rijeka, 51000, Croatia

    Dean Crnković

  4. Sobolev Institute of Mathematics, Novosibirsk, Russia, 630090

    Denis Krotov

  5. CNRS, University of Aix Marseille, Centrale Marseille, I2M, Marseille, France

    Patrick Solé

Authors
  1. Minjia Shi
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  2. Yaya Li
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  3. Wei Cheng
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  4. Dean Crnković
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  5. Denis Krotov
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  6. Patrick Solé
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Corresponding author

Correspondence to Minjia Shi.

Additional information

Communicated by Y. Zhou.

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This work is supported in part by the National Natural Science Foundation of China (Grant No. 12071001). The work of Dean Crnković is supported by Croatian Science Foundation under the project 6732. The work of Denis Krotov is supported within the framework of the state contract of the Sobolev Institute of Mathematics (Project FWNF-2022-0017).

Appendix on complex Hadamard matrices

Appendix on complex Hadamard matrices

In this appendix, we indicate how we constructed the matrices used in our computer experiments. The corresponding complex Hadamard matrices are publicly available on Github.Footnote 2

1.1 Order 8

The matrix is obtained by applying our Proposition 5.

1.2 Order 10

The matrix is obtained by applying our Proposition 8.

1.3 Order 16

One matrix is obtained applying our Theorem 2. The other matrix is obtained from the real Bush-type Hadamard matrix from a matrix in [12], by multiplying all off-diagonal blocks by i.

1.4 Order 18

The matrix is constructed from four Williamson type matrices of order 9 obtained from the database [15] upon using Lemma 6 of [13].

1.5 Order 20

The matrix is obtained by applying our Proposition 5.

1.6 Order 26

The matrix is obtained by applying our Proposition 8.

1.7 Order 32

The matrix is obtained by applying our Proposition 5.

1.8 Order 34

The matrix is constructed from four Williamson type matrices of order 17 obtained from the database [15] upon using Lemma 6 of [13].

1.9 Order 36

One matrix is obtained from the complex Hadamard matrix of order 6 using our Theorem 2. Two matrices are obtained from the real Bush-type Hadamard matrices of order 36 given in [10, 11] by multiplying the off-diagonal blocks with i.

1.10 Order 40

The matrix is obtained by applying our Proposition 5.

1.11 Order 50

One matrix is obtained by applying our Proposition 8. The other matrices are constructed from four Williamson type matrices of order 25 obtained from the database [15] upon using Lemma 6 of [13].

1.12 Order 52

The matrix is obtained by applying our Proposition 5.

1.13 Order 58

The matrix is constructed from four Williamson type matrices of order 29 obtained from the database [15] upon using Lemma 6 of [13].

1.14 Order 64

Two matrices are constructed from two complex Hadamard matrices of order 8 obtained from the database [3] upon using our Theorem 2. Another matrix is obtained from a real Hadamard matrix of order 8, using the proposition 7.

1.15 Order 68

The matrix is obtained by applying our Proposition 5.

1.16 Order 72

Two matrices are obtained by applying our Proposition 5.

1.17 Order 74

Two matrices are constructed from four Williamson type matrices of order 37 obtained from the database [15] upon using Lemma 6 of [13].

1.18 Order 80

The matrix is obtained by applying our Proposition 5.

1.19 Order 82

One matrix is constructed from four Williamson type matrices of order 41 obtained from the database [15] upon using Lemma 6 of [13]. Another matrix is obtained by applying our Proposition 8.

1.20 Order 90

The matrix is constructed from four Williamson type matrices of order 45 obtained from the database [15] upon using Lemma 6 of [13].

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Shi, M., Li, Y., Cheng, W. et al. Self-dual bent sequences for complex Hadamard matrices. Des. Codes Cryptogr. 91, 1453–1474 (2023). https://doi.org/10.1007/s10623-022-01157-6

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