PAF Reconstruction with the Orbits Method

  • Ilias S. Kotsireas
  • Youtong Liu
  • Jing YangEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11661)


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.


Periodic autocorrelation function D-optimal design Orbits method 



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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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Wilfrid Laurier UniversityWaterlooCanada
  2. 2.SMS-HCICGuangxi University for NationalitiesNanningPeople’s Republic of China

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