The Construction of Regular Hadamard Matrices by Cyclotomic Classes

Abstract

For every prime power \(q \equiv 7 \; mod \; 16\), there are (qa, b, c, d)-partitions of GF(q), with odd integers a, b, c, and d, where \(a \equiv \pm 1 \; mod \; 8\) such that \(q=a^2+2(b^2+c^2+d^2)\) and \(d^2=b^2+2ac+2bd\). Many results for the existence of \(4-\{q^2; \; \frac{q(q-1)}{2}; \; q(q-2) \}\) SDS which are simple homogeneous polynomials of parameters a, b, c and d of degree at most 2 have been found. Hence, for each value of q, the construction of SDS becomes equivalent to building a \((q; \; a, \; b, \; c, \; d)\)-partition. Once this is done, the verification of the construction only involves verifying simple conditions on a, b, c and d which can be done manually.

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References

  1. 1.

    Baumert, L.D., Mills, W.H., Ward, R.L.: Uniform cyclotomy. J. Number Theory 14, 67–82 (1982)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Jungnickel, D.: Finite Fields: Structure and Arithmetric. BI-Wissenschaftsverlag, Mannheim (1993)

    Google Scholar 

  3. 3.

    Lang, S.: Algebraic Number Theory. Springer, New York (1986)

    Google Scholar 

  4. 4.

    Leung, K.H., Ma, S.L., Schmidt, B.: New Hadamard matrices of order \(4p^2\) obtained from Jacobi sums of order 16. J. Combin. Theory Ser. A 113, 822–838 (2006)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Lidl, R., Niederreiter, H.: Introduction to Finite Fields and Their Applications. Cambridge University Press, Cambridge (1994)

    Google Scholar 

  6. 6.

    Pott, A.: Finite Geometry and Character Theory. Lecture Notes in Mathematics, vol. 1601. Springer, New York (1995)

    Google Scholar 

  7. 7.

    Xia, M., Liu, G.: An infinite class of supplementary difference sets and Williamson matrices. J. Comb. Theory Ser. A 58, 310–317 (1991)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Xia, M., Liu, G.: A new family of supplementary difference sets and Hadamard matrices. J. Stat. Plan. Inference 51, 283–291 (1996)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Xia, M., Xia, T., Seberry, J.: A new method for constructing Williamson Matrices. J. Des. Codes Cryptogr. 35, 191–209 (2005)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Xia, M., Xia, T., Seberry, J., Zuo, G.: A new method for constructing T-matrices. Aust. J. Combin. 32, 61–78 (2005)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Xia, T., Xia, M., Seberry, J.: Regular Hadamard Matrix, maximum excess and SBIBD. Aust. J. Combin. 27, 263–275 (2003)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Xia, T., Xia, M., Seberry, J.: Some new results of regular Hadamard matrices and SBIBD. Aust. J. Combin. 37, 117–126 (2007)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Xia, T., Seberry, J., Xia, M.: New constructing of regular Hadamard matrices. WSEAS Trans. Math. 9(5), 1068–1073 (2006)

    MathSciNet  Google Scholar 

  14. 14.

    Xiang, Q.: Difference families from line and half line. Eur. J. Combin. 19, 395–400 (1998)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

The authors are grateful to the referees and Prof. Majid Soleimani-damaneh for their helpful comments.

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Correspondence to Tianbing Xia.

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Communicated by Behruz Tayfeh-Rezaie.

Appendices

Appendix A

Table of parameters a, b, c and d in the \((q; a, \; b, \; c, \; d)\)-partitions of GF(q) for \(q<1000\)

q \(k^{*}\) a b c d Applicable theorems
7 2 \(-\)1 1 1 1 Coro. 5.3 (i), (ii), Coro. 6.3 (i)-(vii), Coro. 7.3 (i), (ii)
23 2 1 3 \(-\)1 \(-\)1 Coro.  6.3 (ii), (iii), Coro. 7.3 (iii)
71 8 \(-\)7 1 \(-\)1 \(-\)3 Coro.  6.3 (ii)
103 2 \(-\)7 5 1 \(-\)1 None
151 9 \(-\)1 5 7 \(-\)1 Coro.  6.3 (ii), (vii)
167 2 \(-\)1 3 7 5 Coro. 5.3 (ii),Coro.  6.3 (v)
199 13 \(-\)1 3 9 3 Coro. 5.3 (i), Coro.  6.3 (i)
263 2 9 \(-\)9 \(-\)1 3 Coro.  6.3 (ii)
311 4 7 \(-\)1 7 9 None
343 \(\delta \) + 1 7 11 1 \(-\)5 \(\hbox {None}^{**}\)
359 11 \(-\)9 3 \(-\)7 \(-\)9 Coro.  6.3 (ii)
439 9 7 \(-\)5 1 \(-\)13 Coro. 7.3 (ii)
487 3 \(-\)1 \(-\)5 \(-\)7 \(-\)13 None
503 6 \(-\)17 \(-\)9 \(-\)1 5 None
599 11 \(-\)23 3 \(-\)1 \(-\)5 Coro.  6.3 (ii)
631 5 1 5 \(-\)17 1 Coro.  6.3 (ii)
647 2 \(-\)9 \(-\)15 7 3 None
727 2 \(-\)25 \(-\)5 1 \(-\)5 Coro. 5.3 (i), Coro.  6.3 (i), Coro. 7.3 (iii)
743 2 17 \(-\)13 \(-\)7 \(-\)3 None
823 3 \(-\)7 \(-\)9 9 \(-\)15 None
839 4 17 \(-\)7 \(-\)1 \(-\)15 None
887 2 7 9 \(-\)7 17 None
919 6 17 15 \(-\)9 3 Coro.  6.3 (ii)
967 2 \(-\)17 \(-\)13 \(-\)7 11 None
983 2 \(-\)7 \(-\)3 \(-\)17 13 None
  1. *Generator \(g=\omega + k\).
  2. **\(\delta \) is a generator of GF(343) and satisfies \(\delta ^3=\delta +5\). Regular Hadamard matrices of order \(4.7^{2r}\) have been constructed by [11] for all \(r \ge 1\)

Appendix B. 34 Equivalence Subsets of the Probable Sets

  • \(\mid I \mid = 1\), 1 equivalence subset \(\{ 0 \}\);

  • \(\mid I \mid = 3\), 6 equivalence subsets:

    $$\begin{aligned} \{ 0, 1, 2 \}&\equiv 11 \{ 0, 3, 6 \}, \\ \{ 0, 1, 3 \}&\equiv 13 \{ 0, 1, 6 \} + 3 \equiv 15 \{ 0, 1, 14 \} + 1 \equiv 3 \{ 0, 1, 11 \}, \\ \{ 0, 1, 4 \}&\equiv 3 \{ 0, 1, 5 \} + 1 \equiv 13 \{ 0, 1, 12 \} + 4 \equiv 15 \{ 0, 1, 13 \} + 1, \\ \{ 0, 1, 7 \}&\equiv 9 \{ 0, 1, 10 \} + 7 \equiv 3 \{ 0, 2, 5 \} + 1 \equiv 11 \{ 0, 2, 13 \} +1, \\ \{ 0, 2, 4\}&\equiv 13 \{ 0, 4, 10 \}, \\ \{ 0, 2, 6 \}&\end{aligned}$$
  • \(\mid I \mid = 5\), 17 equivalence subsets:

    $$\begin{aligned} \{ 0, 1, 2, 3, 4 \}&\equiv 3 \{ 0, 1, 5, 6, 11 \} + 1, \\ \{ 0, 1, 2, 3, 5 \}&\equiv 15 \{ 0, 1, 2, 3, 14 \}\\&\quad +3 \equiv 11 \{ 0, 1, 4, 7, 10 \} + 5 \equiv 3 \{ 0, 1, 5, 10, 11 \}+2, \\ \{ 0, 1, 2, 3, 6 \}&\equiv 15 \{ 0, 1, 2, 3, 13 \} + 3 \equiv 3 \{ 0, 1, 2, 6, 11 \} \equiv 13 \{ 0, 1, 2, 7, 12 \} + 6, \\ \{ 0, 1, 2, 3, 7 \}&\equiv 15 \{ 0, 1,2,3,12 \}+3 \equiv 5 \{0, 1, 3, 6, 13 \}+2 \equiv 13 \{ 0, 1, 4, 6, 11 \} + 3, \\ \{ 0, 1, 2, 4, 5 \}&\equiv 15 \{ 0, 1, 2, 13, 14 \} + 2 \equiv 11 \{ 0, 1, 4, 7, 13 \} + 5 \equiv 5 \{ 0, 1, 4, 10, 13 \}, \\ \{ 0, 1, 2, 4, 6 \}&\equiv 15 \{ 0, 1, 2, 12, 14 \} + 2 \equiv 5 \{ 0, 1, 3, 7, 13 \} + 1 \equiv 11 \{ 0, 1, 4, 10, 14 \}+6, \\ \{ 0, 1, 2, 4, 7 \}&\equiv 3 \{ 0, 1, 2, 5, 11 \}+1 \equiv 13 \{ 0, 1, 2, 7, 13 \} + 7 \equiv 15 \{ 0, 1, 2, 11, 14\} +2, \\ \{ 0, 1, 2, 4, 11 \}&\equiv 9 \{ 0, 1, 2, 7, 14 \} + 2 \equiv 11 \{ 0, 1, 3, 6, 12 \} \equiv 3 \{ 0, 1, 5, 11, 14 \}+1, \\ \{ 0, 1, 2, 4, 13 \}&\equiv 15 \{ 0, 1, 2, 5, 14 \} + 2 \equiv 13 \{ 0, 1, 4, 5, 10 \} \equiv 3 \{ 0, 1, 4, 5, 11 \} + 1, \\ \{ 0, 1, 2, 4, 14 \}&\equiv 13 \{ 0, 1, 2, 6, 12\} + 4, \\ \{ 0, 1, 2, 5, 6 \}&\equiv 15 \{ 0, 1, 2, 12, 13 \} + 2 \equiv 11 \{ 0, 1, 3, 4, 7 \}+5 \equiv 5\{ 0, 1, 3, 4, 13 \} + 1, \\ \{ 0, 1, 2, 5, 7 \}&\equiv 15 \{ 0, 1, 2, 11, 13 \} + 2\equiv 5 \{ 0, 1, 3, 6, 7 \}+2 \equiv 11 \{ 0, 1, 4, 6, 7 \} +5, \\ \{ 0, 1, 2, 5, 12 \}&\equiv 15 \{ 0, 1, 2, 6, 13 \} + 2 \equiv 13 \{ 0, 1, 3, 7, 12 \} +5 \equiv 5\{ 0, 1, 4, 5, 14 \} -4, \\ \{ 0, 1, 2, 6, 7 \}&\equiv 15 \{ 0, 1, 2, 11, 12 \} + 2 \equiv 5 \{ 0, 1, 3, 4, 6 \} + 2 \equiv 11 \{ 0, 1, 3, 4, 14 \} +6, \\ \{ 0, 1, 2, 7, 11 \}&\equiv 11 \{ 0, 1, 3, 5, 6 \}), \\ \{ 0, 1, 3, 5, 7 \}&\equiv 11 \{ 0, 1, 4, 6, 10 \} + 5 \equiv 3 \{ 0, 1, 7, 11, 13 \} \equiv 9 \{ 0, 1, 10, 12, 14 \} +7, \\ \{ 0, 1, 3, 5, 14 \}&\equiv 9 \{ 0, 1, 3, 12, 14 \} + 5\equiv 11 \{ 0, 1, 4, 10, 11 \} + 5 \equiv 3\{ 0, 1, 6, 7, 13 \} -2; \end{aligned}$$
  • \(\mid I \mid = 7\), 10 equivalence subsets:

    $$\begin{aligned} \{0, 1, 2, 3, 4, 5, 6\}&\equiv 3 \{0, 1, 2, 6, 7, 11, 12\}), \\ \{0, 1, 2, 3, 4, 5, 7\}&\equiv 15 \{0, 1, 2, 3, 4, 5, 14\}+5\equiv 3 \{0, 1, 2, 5, 6, 11, 12 \}+1 \\&\equiv 13 \{0, 1, 2, 6, 7, 12, 13\}+7, \\ \{0, 1, 2, 3, 4, 6, 7\}&\equiv 15 \{0, 1, 2, 3, 4, 13, 14\}+4\equiv 3 \{0, 1, 2, 5, 6, 7, 11 \}+1 \\&\equiv 13 \{0, 1, 2, 5, 6, 7, 12\}+6, \\ \{0, 1, 2, 3, 4, 6, 13\}&\equiv 15 \{0, 1, 2, 3, 4, 7, 14\}+4\equiv 13 \{0, 1, 2, 3, 6, 7, 12 \}+6 \\&\equiv 3 \{0, 1, 2, 3, 7, 12, 13\}-3, \\ \{0, 1, 2, 3, 4, 7, 13\}&\equiv 3 \{0, 1, 2, 4, 5, 6, 11\}+1, \\ \{0, 1, 2, 3, 5, 6, 7\}&\equiv 15 \{0, 1, 2, 3, 12, 13, 14\}+3\equiv 13 \{0, 1, 2, 5, 7, 11, 12 \}+6 \\&\equiv 3 \{0, 1, 2, 6, 7, 11, 13\}, \\ \{0, 1, 2, 3, 5, 6, 12\}&\equiv 15 \{0, 1, 2, 3, 7, 13, 14\}+3\equiv 5 \{0, 1, 2, 4, 5, 11, 14 \}-4 \\&\equiv 11 \{0, 1, 2, 4, 7, 13, 14\}+6, \\ \{0, 1, 2, 3, 5, 7, 12\}&\equiv 5 \{0, 1, 2, 3, 6, 7, 13\}+2\equiv 11 \{0, 1, 2, 3, 6, 12, 13 \}+1 \\&\equiv 15 \{0, 1, 2, 3, 7, 12, 14\}+3, \\ \{0, 1, 2, 3, 5, 7, 14\}&\equiv 9 \{0, 1, 2, 3, 5, 12, 14\}+5\equiv 3 \{0, 1, 3, 6, 7, 12, 13 \}-2 \\&\equiv 11 \{0, 1, 4, 5, 10, 12, 14\}+7, \\ \{0, 1, 2, 4, 5, 7, 11\}&\equiv 11 \{0, 1, 2, 4, 6, 7, 13\}+5\equiv 3 \{0, 1, 2, 5, 11, 12, 14 \}+1 \\&\equiv 9 \{0, 1, 2, 7, 11, 13, 14\}+2. \end{aligned}$$

To simplify the notation we omit “\(mod \; 16\)”, and for each equivalence class we only enumerate the probable sets, which occupy positions at the head in lexicographic order from 0 to 15, and the remainder unlisted in the class, might be represented as \(I+k\) or \(qI+k\) with an integer k and an enumerated probable set I in these classes.

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Xia, T., Xia, M. & Seberry, J. The Construction of Regular Hadamard Matrices by Cyclotomic Classes. Bull. Iran. Math. Soc. (2020). https://doi.org/10.1007/s41980-020-00402-9

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Keywords

  • Regular Hadamard matrix
  • Cyclotomic class
  • Partition
  • Supplement difference sets (SDS)

Mathematics Subject Classification

  • 05A18
  • 05B10
  • 05B20