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PAF Reconstruction with the Orbits Method

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Computer Algebra in Scientific Computing (CASC 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11661))

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Abstract

The maximal determinant problem for \(-1/{+1}\) matrices has been studied extensively and upper bounds for the determinant are known for various classes of matrices. These upper bounds are attained by specific kinds of combinatorial matrices and D-optimal matrices are one such case. One of the key issues in the search for D-optimal matrices is to reconstruct a \(-1/{+1}\) sequence of length n from a given sequence of periodic autocorrelation function (PAF) values. In turn, this is reduced to solving a quadratic system with \(\lfloor n/2\rfloor \) equations over \(\{-1,{+1}\}^n\). In this paper, a method for reconstructing a special class of \(-1/{+1}\) sequences is proposed by making use of some combinatorial properties of PAF values and the orbits method based on group actions. Furthermore, we apply additional filtering criteria to enhance the effectiveness of the method. Experiments show that the new approach can solve relatively large-scale problems and can help to generate solutions for many D-optimal problems.

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Notes

  1. 1.

    Obviously, \(p_0=n\). It is omitted unless it is used.

  2. 2.

    If \(n_1>1\) and \(n_t=1\), \((n_t,n_1)\) is regarded as a fragment of the form (1, k).

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Acknowledgements

This work was made possible by the facilities of the CARGO Lab at Wilfrid Laurier University, and the SMS International as well as the Key Laboratory of Software Engineering (2018-18XJSY-03) at Guangxi University for Nationalities. ISK’s work is supported by NSERC grants. JY and YL’s work is supported by NSFC (No. 10801101), the Special Fund for Guangxi Bagui Scholars, Guangxi Science and Technology Program (AD18126010), and the Startup Foundation for Advanced Talents in Guangxi University for Nationalities (2015MDQD018).

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Authors and Affiliations

  1. Wilfrid Laurier University, Waterloo, ON, N2L 3C5, Canada

    Ilias S. Kotsireas

  2. SMS-HCIC, Guangxi University for Nationalities, Nanning, 530006, People’s Republic of China

    Youtong Liu & Jing Yang

Authors
  1. Ilias S. Kotsireas
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  2. Youtong Liu
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  3. Jing Yang
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Corresponding author

Correspondence to Jing Yang .

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Editors and Affiliations

  1. Coventry University, Coventry, UK

    Matthew England

  2. University of Kassel, Kassel, Germany

    Wolfram Koepf

  3. Plekhanov Russian University of Economics, Moscow, Russia

    Timur M. Sadykov

  4. University of Kassel, Kassel, Germany

    Werner M. Seiler

  5. Russian Academy of Sciences, Novosibirsk, Russia

    Evgenii V. Vorozhtsov

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Kotsireas, I.S., Liu, Y., Yang, J. (2019). PAF Reconstruction with the Orbits Method. In: England, M., Koepf, W., Sadykov, T., Seiler, W., Vorozhtsov, E. (eds) Computer Algebra in Scientific Computing. CASC 2019. Lecture Notes in Computer Science(), vol 11661. Springer, Cham. https://doi.org/10.1007/978-3-030-26831-2_21

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  • DOI: https://doi.org/10.1007/978-3-030-26831-2_21

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