Annals of Combinatorics

, 13:305 | Cite as

Weighing Matrices and String Sorting

  • Ilias S. Kotsireas
  • Christos Koukouvinos
  • Jennifer Seberry
Article

Abstract

In this paper we establish a fundamental link between the search for weighing matrices constructed from two circulants and the operation of sorting strings, an operation that has been studied extensively in computer science. In particular, we demonstrate that the search for weighing matrices constructed from two circulants using the power spectral density criterion and exploiting structural patterns for the locations of the zeros in candidate solutions, can be viewed as a string sorting problem together with a linear time algorithm to locate common strings in two sorted arrays. This allows us to bring into bear efficient algorithms from the string sorting literature. We also state and prove some new enhancements to the power spectral density criterion, that allow us to treat successfully the rounding error effect and speed up the algorithm. Finally, we use these ideas to find new weighing matrices of order 2n and weights 2n – 13, 2n – 17 constructed from two circulants.

AMS Subject Classification

05B20 62K05 

Keywords

weighing matrices algorithm pattern locations of zeros power spectral density rounding error 

References

  1. 1.
    Arasu K.T., Gulliver T.A.: Self-dual codes over \({{\mathbb {F}}_p}\) and weighing matrices. IEEE Trans. Inform. Theory 47, 2051–2055 (2001)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Craigen R.: Weighing matrices and conference matrices. In: Colbourn, C.J., Dinitz, J.H. (eds) The CRC Handbook of Combinatorial Designs, pp. 496–504. CRC Press, Boca Raton (1996)Google Scholar
  3. 3.
    Craigen R.: The structure of weighing matrices having large weights. Des. Codes Cryptogr. 5, 199–216 (1995)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Craigen R., Kharaghani H.: Orthogonal designs. In: Colbourn, C.J., Dinitz, J.H. (eds) The CRC Handbook of Combinatorial Designs, pp. 280–295. CRC Press, Boca Raton (2006)Google Scholar
  5. 5.
    Fletcher R.J., Gysin M., Seberry J.: Application of the discrete Fourier transform to the search for generalised Legendre pairs and Hadamard matrices. Australas. J. Combin. 23, 75–86 (2001)MATHMathSciNetGoogle Scholar
  6. 6.
    Geramita A.V., Seberry J.: Orthogonal Designs: Quadratic Forms and Hadamard Matrices. Marcel Dekker Inc., New York (1979)MATHGoogle Scholar
  7. 7.
    Knuth, D.E.: The Art of Computer Programming, Vol. 3: Sorting and Searching, Addison- Wesley Series in Computer Science and Information Processing. Addison-Wesley Publishing Co., Mass.- London-Don Mills (1973)Google Scholar
  8. 8.
    Koukouvinos C., Seberry J.: New weighing matrices and orthogonal designs constructed using two sequences with zero autocorrelation function-a review. J. Statist. Plann. Inference 81, 153–182 (1999)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Kreher D.L., Stinson D.R.: Combinatorial Algorithms: Generation, Enumeration and Search. CRC press, Boca Raton (1998)Google Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  • Ilias S. Kotsireas
    • 1
  • Christos Koukouvinos
    • 2
  • Jennifer Seberry
    • 3
  1. 1.Department of Physics and Computer ScienceWilfrid Laurier UniversityWaterlooCanada
  2. 2.Department of MathematicsNational Technical University of AthensAthensGreece
  3. 3.Centre for Computer Security Research, School of Information Technology and Computer ScienceUniversity of WollongongWollongongAustralia

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