A feasibility approach for constructing combinatorial designs of circulant type

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.

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Fig. 1

Notes

  1. 1.

    The support of \(c=(c_0,c_1,\ldots ,c_{n-1})\in {\mathbb {R}}^n\) is the set \(\{i\in \{0,\ldots ,n-1\}: c_i\ne 0\}\).

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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.

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Correspondence to Matthew K. Tam.

Additional information

Dedicated to Jonathan M. Borwein.

This work is dedicated to the late Jonathan M. Borwein who suggested this project during his 2016 sabbatical in Canada. FJAA and RC were partially supported by Ministerio de Economía, Industria y Competitividad (MINECO) and European Regional Development Fund (ERDF), Grant MTM2014-59179-C2-1-P. FJAA was supported by the Ramón y Cajal program by MINECO and ERDF (RYC-2013-13327) and RC was supported by MINECO and European Social Fund (BES-2015-073360) under the program “Ayudas para contratos predoctorales para la formación de doctores 2015”. IK is supported by an Natural Sciences and Engineering Research Council Grant. MKT was supported by Deutsche Forschungsgemeinschaft RTG2088 and by a Postdoctoral Fellowship from the Alexander von Humboldt Foundation.

Appendix: Detailed results for CW matrices

Appendix: Detailed results for CW matrices

Table 3 Results for CW matrices (10 random initialization, 3600 s time limit)

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Aragón Artacho, F.J., Campoy, R., Kotsireas, I. et al. A feasibility approach for constructing combinatorial designs of circulant type. J Comb Optim 35, 1061–1085 (2018). https://doi.org/10.1007/s10878-018-0250-5

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Keywords

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