Skip to main content

Goethals–Seidel Difference Families with Symmetric or Skew Base Blocks


We single out a class of difference families which is widely used in some constructions of Hadamard matrices and which we call Goethals–Seidel (GS) difference families. They consist of four subsets (base blocks) of a finite abelian group of order v, which can be used to construct Hadamard matrices via the well-known Goethals–Seidel array. We consider the special class of these families in cyclic groups, where each base block is either symmetric or skew. We omit the well-known case where all four blocks are symmetric. By extending previous computations by several authors, we complete the classification of GS-difference families of this type for odd \(v<50\). In particular, we have constructed the first examples of so called good matrices, G-matrices and best matrices of order 43, and good matrices and G-matrices of order 45. We also point out some errors in one of the cited references.

This is a preview of subscription content, access via your institution.


  1. Di Mateo, O., Đoković, D.Ž., Kotsireas, I.S.: Symmetric Hadamard matrices of order 116 and 172 exist. Spec. Matrices 3, 227–234 (2015)

    MathSciNet  MATH  Google Scholar 

  2. Đoković, D.Ž.: Good matrices of orders 33, 35 and 127. JCMCC 14, 145–152 (1993)

    MathSciNet  MATH  Google Scholar 

  3. Đoković, D.Ž.: Supplementary difference sets with symmetry for Hadamard matrices. Oper. Matrices 3, 557–569 (2009)

    MathSciNet  Article  Google Scholar 

  4. Duke, W.: Some old problems and new results about quadratic forms. Not. Am. Math. Soc. 44, 190–196 (1997)

    MathSciNet  MATH  Google Scholar 

  5. Fletcher, R.J., Gysin, M., Seberry, J.: pplication of the discrete Fourier transform to the search for generalised Legendre pairs and Hadamard matrices. Australas. J. Combin. 23, 75–86 (2001)

    MathSciNet  MATH  Google Scholar 

  6. Georgiou, S., Koukouvinos, C.: On circulant G-matrices. JCMCC 40, 205–225 (2002)

    MATH  Google Scholar 

  7. Georgiou, S., Koukouvinos, C., Seberry, J.: On circulant best matrices and their applications. Linear Multilinear Algebra 48, 263–274 (2001)

    MathSciNet  Article  Google Scholar 

  8. Georgiou, S., Koukouvinos, C., Stylianou, S.: On good matrices, skew Hadamard matrices and optimal designs. Comput. Stat. Data Anal. 41, 171–184 (2002)

    MathSciNet  Article  Google Scholar 

  9. Holzmann, W.H., Kharaghani, H., Tayfeh-Rezaie, B.: Williamson matrices up to order 59. Des. Codes Cryptogr. 46, 343–352 (2008)

    MathSciNet  Article  Google Scholar 

  10. Koukouvinos, C., Seberry, J.: On G-matrices. Bull. Inst. Combin. Appl. 9, 40–44 (1993)

    MathSciNet  MATH  Google Scholar 

  11. Pearl, J.: Heuristics: intelligent search strategies for computer problem solving. Addison-Wesley, Boston (1984)

    Google Scholar 

  12. Seberry Wallis, J.: On Hadamard matrices. J. Combin. Theory 18, 149–164 (1975)

    Article  Google Scholar 

  13. Seberry, J., Yamada, M.: Hadamard matrices, sequences, and block designs. In: Contemporary design theory. Wiley-Interscience Series in Discrete Mathematics and Optimization, Wiley, New York, pp. 431–560 (1992)

  14. Spence, E.: Skew Hadamard matrices of order \(2(q+1)\). Discrete Math. 18, 79–85 (1977)

    MathSciNet  Article  Google Scholar 

  15. Szekeres, G.: A note on skew type orthogonal \({\pm } 1\) matrices. In: Hajnal, A., Lovasz, L., Sos, V.T. (eds.) Combinatorics, Colloquia Mathematica Societatis Janos Bolyai, vol. 52, pp. 489–498. North Holland, Amsterdam (1989)

    Google Scholar 

  16. Wallis, J.S.: Hadamard matrices. In: LNM 292, Combinatorics: Room Squares, Sum-Free Sets, Hadamard matrices. Springer, Berlin (1972)

    Google Scholar 

  17. Xia, M., Xia, T., Seberry, J., Wu, J.: An infinite series of Goethals–Seidel arrays. Discrete Appl. Math. 145, 498–504 (2005)

    MathSciNet  Article  Google Scholar 

  18. Zhang, X.M.: G-matrices of order 19. Bull. Inst. Combin. Appl. 4, 95–98 (1992)

    MathSciNet  MATH  Google Scholar 

Download references


The authors would like to thank the referees for their constructive comments, that contributed to improving the paper. The authors wish to acknowledge generous support by NSERC, Grant Numbers 5285-2012 and 213992. Computations were performed on the SOSCIP Consortium’s Blue Gene/Q, computing platform. SOSCIP is funded by the Federal Economic Development Agency of Southern Ontario, IBM Canada Ltd., Ontario Centres of Excellence, Mitacs and 14 academic member institutions.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Dragomir Ž. Ɖoković.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Ɖoković, D.Ž., Kotsireas, I.S. Goethals–Seidel Difference Families with Symmetric or Skew Base Blocks. Math.Comput.Sci. 12, 373–388 (2018).

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • Goethals–Seidel array
  • Difference families
  • Good matrices
  • G-matrices
  • Best matrices

Mathematics Subject Classification

  • 05B10
  • 05B20