A feasibility approach for constructing combinatorial designs of circulant type

  • Francisco J. Aragón Artacho
  • Rubén Campoy
  • Ilias Kotsireas
  • Matthew K. Tam
Article
  • 32 Downloads

Abstract

In this work, we propose an optimization approach for constructing various classes of circulant combinatorial designs that can be defined in terms of autocorrelation. The problem is formulated as a so-called feasibility problem having three sets, to which the Douglas–Rachford projection algorithm is applied. The approach is illustrated on three different classes of circulant combinatorial designs: circulant weighing matrices, D-optimal matrices of circulant type, and Hadamard matrices with two circulant cores. Furthermore, we explicitly construct two new circulant weighing matrices, a CW(126, 64) and a CW(198, 100), whose existence was previously marked as unresolved in the most recent version of Strassler’s table.

Keywords

Strassler’s table Circulant weighing matrices Circulant combinatorial designs Douglas–Rachford algorithm 

Notes

Acknowledgements

We would like to thank an anonymous referee for their careful reading and suggestions, which helped us to improve the manuscript. We would also like to thank Yossi Strassler for sending us a scan of his original table. This research was enabled, in part, thanks to support provided by Compute Canada and their Graham petascale supercomputer.

References

  1. Aragón Artacho FJ, Borwein JM, Tam MK (2014a) Recent results on Douglas-Rachford methods for combinatorial optimization problems. J Optim Theory Appl 163(1):1–30Google Scholar
  2. Aragón Artacho FJ, Borwein JM, Tam MK (2014b) Douglas-Rachford feasibility methods for matrix completion problems. ANZIAM J 55(4):299–326Google Scholar
  3. Aragón Artacho FJ, Borwein JM, Tam MK (2016) Global behavior of the Douglas-Rachford method for a nonconvex feasibility problem. J Glob Optim 65(2):309–327MathSciNetCrossRefMATHGoogle Scholar
  4. Aragón Artacho FJ, Campoy R (accepted Nov. 2017) Solving graph coloring problems with the Douglas–Rachford algorithm. Set-Valued Var. Anal., p 27.  https://doi.org/10.1007/s11228-017-0461-4
  5. Arasu KT, Dillon JF (1999) Difference sets. Sequences and their correlation properties. In: Pott A, Kumar PV, Helleseth T, Jungnickel D (eds) Perfect ternary arrays. Springer, Dordrecht, pp 1–15Google Scholar
  6. Arasu KT, Gulliver TA (2001) Self-dual codes over \({\mathbb{F}}_p\) and weighing matrices. IEEE Trans Inf Theory 47(5):2051–2055CrossRefMATHGoogle Scholar
  7. Arasu KT, Gutman AJ (2010) Circulant weighing matrices. Cryptogr Commun 2:155–171CrossRefGoogle Scholar
  8. Arasu KT, Kotsireas IS, Koukouvinos C, Seberry J (2010) On circulant and two-circulant weighing matrices. Australas J Combin 48:43–51MathSciNetMATHGoogle Scholar
  9. Arasu KT, Leung KH, Ma SL, Nabavi A, Ray-Chaudhuri DK (2006a) Circulant weighing matrices of weight \(2^{2t}\). Des Codes Cryptogr 41(1):111–123Google Scholar
  10. Arasu KT, Leung KH, Ma SL, Nabavi A, Ray-Chaudhuri DK (2006b) Determination of all possible orders of weight 16 circulant weighing matrices. Finite Fields Appl 12(4):498–538Google Scholar
  11. Bauschke HH, Combettes PL (2011) Convex analysis and monotone operator theory in hilbert spaces. Springer, New YorkCrossRefMATHGoogle Scholar
  12. Bauschke HH, Combettes PL, Luke DR (2004) Finding best approximation pairs relative to two closed convex sets in Hilbert space. J Approx Theory 127(2):178–192MathSciNetCrossRefMATHGoogle Scholar
  13. Bauschke HH, Dao MN (2017) On the finite convergence of the Douglas-Rachford algorithm for solving (not necessarily convex) feasibility problems in Euclidean spaces. SIAM J Optim 27:207–537MathSciNetCrossRefMATHGoogle Scholar
  14. Borwein JM, Lewis AS (2006) Convex analysis and nonlinear optimization. Springer, New YorkCrossRefMATHGoogle Scholar
  15. Brent RP (2013) Finding D-optimal design by randomised decomposition and switching. Australas J Combin 55:15–30MathSciNetMATHGoogle Scholar
  16. Briggs WL, Henson VE (1995) DFT. An owner’s manual for the discrete Fourier transform, SIAM, PhiladelphiaMATHGoogle Scholar
  17. Cohn JHE (1989) On determinants with elements \(\pm \)1. Bull Lond Math Soc 21(1):36–42MathSciNetCrossRefMATHGoogle Scholar
  18. Colbourn CJ, Dinitz JH (2007) Handbook of combinatorial designs, 2nd edn. Chapman & Hall, Boca RatonMATHGoogle Scholar
  19. Đoković DZ̆, Kotsireas IS (2012) New results on D-pptimal matrices. J Combin Des 20(6):278–289Google Scholar
  20. Đoković DZ̆, Kotsireas IS (2015a) Compression of periodic complementary sequences and applications. Des Codes Cryptogr 74(2):365–377Google Scholar
  21. Đoković DZ̆, Kotsireas IS (2015b) D-optimal matrices of orders 118, 138, 150, 154 and 174. In: Colbourn CJ (ed) Algebraic design theory and Hadamard matrices. Springer, Basel, pp 71–82Google Scholar
  22. Ehlich H (1964) Determinantenabschätzungen für binäre Matrizen. Math Zeitschr 83:123–132MathSciNetCrossRefGoogle Scholar
  23. Elser V, Rankenburg I, Thibault P (2007) Searching with iterated maps. Proc Natl Acad Sci 104(2):418–426MathSciNetCrossRefMATHGoogle Scholar
  24. Flammia ST, Severini S (2009) Weighing matrices and optical quantum computing. J Phys A 42(6):065302MathSciNetCrossRefMATHGoogle Scholar
  25. Golomb SW, Gong G (2004) Signal design for good correlation. Cambridge University Press, New YorkMATHGoogle Scholar
  26. Gravel S, Elser V (2008) Divide and concur: a general approach to constraint satisfaction. Phys Rev E 78(3):036706CrossRefGoogle Scholar
  27. Gutman AJ (2009) Circulant weighing matrices. Master’s Thesis, Wright State University. http://rave.ohiolink.edu/etdc/view?acc_num=wright1244468669
  28. Hesse R (2014) Fixed point algorithms for nonconvex feasibility with applications. Ph.D. thesis, University of Göttingen. http://hdl.handle.net/11858/00-1735-0000-0022-5F3F-E
  29. Horadam KJ (2012) Hadamard matrices and their applications. Princeton University Press, New JerseyMATHGoogle Scholar
  30. Kotsireas IS (2013) Algorithms and metaheuristics for combinatorial matrices. In: Pardalos PM, Du D-Z, Graham RL (eds) Handbook of combinatorial optimization. Springer, New York, pp 283–309CrossRefGoogle Scholar
  31. Kotsireas IS, Koukouvinos C, Seberry J (2006) Hadamard ideals and Hadamard matrices with two circulant cores. Eur J Combin 27(5):658–668MathSciNetCrossRefMATHGoogle Scholar
  32. Pierra G (1984) Decomposition through formalization in a product space. Math Program 28:96–115MathSciNetCrossRefMATHGoogle Scholar
  33. Sala M, Sakata S, Mora T, Traverso C, Perret L (2009) Gröbner bases, coding, and cryptography. Springer, BerlinCrossRefMATHGoogle Scholar
  34. Seberry JR (2017) Orthogonal designs: hadamard matrices, quadratic forms and algebras. Springer, BerlinCrossRefMATHGoogle Scholar
  35. Seberry J, Yamada M (1992) Hadamard matrices, sequences, and block designs. In: Dintz JH, Stinson DR (eds) Contemporary design theory: a collection of surveys. Wiley, Hoboken, pp 431–560Google Scholar
  36. Strassler Y (1997) The Classification of Circulant Weighing Matrices of Weight 9. Ph.D. thesis, Bar-Ilan University (Israel)Google Scholar
  37. Stinson DR (2004) Combinatorial designs. Constructions and analysis. Springer, New YorkMATHGoogle Scholar
  38. Sturmfels B (2008) Algorithms in Invariant theory. Springer, ViennaMATHGoogle Scholar
  39. Tan MM (2016) Group invariant weighing matrices. arXiv:1610.01914
  40. Tan MM (2014) Relative difference sets and circulant weighing matrices. Ph.D. thesis, Nanyang Technological University. https://repository.ntu.edu.sg/handle/10356/62325
  41. van Dam W (2002) Quantum algorithms for weighing matrices and quadratic residues. Algorithmica 34(4):413–428MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Francisco J. Aragón Artacho
    • 1
  • Rubén Campoy
    • 1
  • Ilias Kotsireas
    • 2
  • Matthew K. Tam
    • 3
  1. 1.Department of MathematicsUniversity of AlicanteAlicanteSpain
  2. 2.CARGO LabWilfrid Laurier UniversityWaterlooCanada
  3. 3.Institut für Numerische und Angewandte MathematikUniversität GöttingenGöttingenGermany

Personalised recommendations