qomo-cheng / hadamard_bent

This repo is for presenting those self-dual bent sequences in various Hadamard matrices

Self-dual bent sequences in Hadamard matrices

This repo is created for presenting various Hadamard matrices and corresponding numerical solutions of self-dual bent seuqnces. It is the open-source of the paper Self-dual Hadamard bent sequences, which has been submitted to IEEE Transactions on XXX.

• Special thanks to Prof. Patrick Solé and Prof. Dean Crnković for constructing and sharing those Hadamard matrices.

Hadamard matrices at different orders

*Note 1: SD for self-dual; seqs for sequences; Refs for references

*Note 2: All Hadamard matrices are under directory: Hadamard_matrices.

*Note 3: All SD bent sequences are under directory: bent_sequences.

Order 16

• Sylvester type
• Number of matrices: 1 link.
• Number of SD bent seqs: 140 link.
• Refs:
  • The first Hadamard matrix of order 16 in Magma

Order 36

• Bush Type
  • Number of matrices: 29 link.
  • Number of SD bent seqs: 64 link.
  • Refs:
    • Z. Janko, "The existence of a Bush-type Hadamard matrix of order 36 and two new infinite classes of symmetric designs," J. Comb. Theory, Ser. A, vol. 95, no. 2, pp. 360-364, Aug. 2001.
    • Z. Janko and H. Kharaghani, "A block negacyclic Bush-type Hadamard matrix and two strongly regular graphs," J. Comb. Theory, Ser. A, vol. 98, no. 1, pp. 118-126, Apr. 2002, DOI: 10.1006/jcta.2001.3231.
• Paley Type
  • Number of matrices: 1 link.
  • Number of SD bent seqs: 204 link.
  • Refs:
    • P. Solé, W. Cheng, S. Guilley, and O. Rioul, "Bent sequences over Hadamard codes for physically unclonable functions," in IEEE International Symposium on Information Theory, Melbourne, Australia, July 12-20, 2021. IEEE, 2021, pp. 801-806, DOI: 10.1109/ISIT45174.2021.9517752.

Order 64

• Regular type

  • Number of matrices: 16 link.
  • Number of SD bent seqs: 2, 4, 6, 12, or 620 link.
  • Refs:
    • D. Crnković, and M.-O. Pavčević, "Some new symmetric designs with parameters (64, 28, 12)," Discrete Math., vol. 237, no. 1-3, pp. 109-118, June 2001.

• Regular type by switching

  • Number of matrices: 1 link.
  • Number of SD bent seqs: 2 link.
  • Refs:
    • D. Crnković, and M.-O. Pavčević, "Some new symmetric designs with parameters (64, 28, 12)," Discrete Math., vol. 237, no. 1-3, pp. 109-118, June 2001.

Order 100

• Regular type

  • Number of matrices: 120 link.
  • Number of SD bent seqs: 1024, 1056, 1152, 1216, 1312, 1536, 1986, 2304, 2336, 2432, 2496, 2576, 2592, 3584, 3712, 3616, 5312, or 6464 download.
  • Refs:
    • D. Crnković, R. Egan, and A. Švob, "Orbit matrices of Hadamard matrices and related codes," Discrete Math., vol. 341, no. 5, pp. 1199-1209, May 2018.

• Regular type

  • Number of matrices: 1 link.
  • Number of SD bent seqs: 12 link.
  • Refs:
    • M.-O. Pavčević, "Symmetric designs of Menon series admitting an action of Frobenius groups," Glas. Mat., III. Ser., vol. 31, no. 2, pp. 209-223, Dec. 1996. [Online]. Available: http://books.google.com/books?id=wdgsTPYo92YC&pg=PA209

• Regular Menon type

  • Number of matrices: 4 link.
  • Number of SD bent seqs: 924 link.
  • Refs:
    • D. Crnković, R. Egan, and A. Švob, "Orbit matrices of Hadamard matrices and related codes," Discrete Math., vol. 341, no. 5, pp. 1199-1209, May 2018.

Order 144

• Bush type
• Number of matrices: 4 link.
• Number of SD bent seqs: 20, 924, or 1052 link.
• Refs:
  • D. Crnković, "A construction of some symmetric (144; 66; 30) designs," J. Appl. Algebra Discrete Struct., vol. 5, no. 1, pp. 33-39, 2007.
  • M.-O. Pavčević, "Symmetric designs of Menon series admitting an action of Frobenius groups," Glas. Mat., III. Ser., vol. 31, no. 2, pp. 209-223, Dec. 1996. [Online]. Available: http://books.google.com/books?id=wdgsTPYo92YC&pg=PA209

Order 196

• Regular type
• Number of matrices: 4 link.
• Number of SD bent seqs: 6864 or 12870 link.
• Refs:
  • D. Crnković, "A construction of some symmetric designs with parameters (196; 91; 42)," Int. Math. Forum, vol. 2, no. 61-64, pp. 3021-3026, 2007.

Copyright and License

This repository is placed into the public domain. Anyone can redistribute it and/or modify it under the terms of the GNU General Public License version 3.0.

Copyright (C) 2022. All Rights Reserved to Authors.