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Energy bounds for codes in polynomial metric spaces

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Abstract

In this article we present a unified treatment for obtaining bounds on the potential energy of codes in the general context of polynomial metric spaces (PM-spaces). The lower bounds we derive via the linear programming techniques of Delsarte and Levenshtein are universally optimal in the sense that they apply to a broad class of energy functionals and, in general, cannot be improved for the specific subspace. Tests are presented for determining whether these universal lower bounds (ULB) can be improved on larger spaces. Our ULBs are applicable on the Euclidean sphere, infinite projective spaces, as well as Hamming and Johnson spaces. Asymptotic results for the ULB for the Euclidean spheres and the binary Hamming space are derived for the case when the cardinality and dimension of the space grow large in a related way. Our results emphasize the common features of the Levenshtein’s universal upper bounds for the cardinality of codes with given separation and our ULBs for energy. We also introduce upper bounds for the energy of designs in PM-spaces and the energy of codes with given separation.

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Notes

  1. For recent generalizations of Gegenbauer polynomials. see [34]

  2. A finite PM-space \(\mathcal {M}\) is called decomposable [40] if there exist a positive integer \(\ell \ge 2\) and metric subspaces \(\mathcal {M}_i\), \(i=1,\ldots ,\ell \), of \(\mathcal {M}\), such that the following three conditions are satisfied: \(\mathcal {M}=\cup _{i=1}^\ell \mathcal {M}_i\), each \(\mathcal {M}_i\) is isometric to the same \(\widetilde{\mathcal {M}}\) with the same standard substitution \(\sigma \), and for any \(x,y \in \mathcal {M}\) the number of the subspaces \(\mathcal {M}_i\) containing simultaneously x and y is equal to \(\frac{\ell |\widetilde{\mathcal {M}}|(\sigma (d(x,y)+1)}{2|\mathcal {M}|}\).

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Acknowledgements

Research for this article was conducted while the authors were in residence at the Institute for Computational and Experimental Research in Mathematics in Providence, RI, during the “Point Configurations in Geometry, Physics and Computer Science” program supported by the National Science Foundation under Grant No. DMS-1439786.

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Authors and Affiliations

  1. Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 8 G Bonchev Str., 1113, Sofia, Bulgaria

    P. G. Boyvalenkov

  2. Technical Faculty, South-Western University, Blagoevgrad, Bulgaria

    P. G. Boyvalenkov

  3. Department of Mathematical Sciences, Purdue University Fort Wayne, Fort Wayne, IN, 46805, USA

    P. D. Dragnev

  4. Center for Constructive Approximation, Department of Mathematics, Vanderbilt University, Nashville, TN, 37240, USA

    D. P. Hardin & E. B. Saff

  5. Faculty of Mathematics and Informatics, Sofia University, 5 James Bourchier Blvd., 1164, Sofia, Bulgaria

    M. M. Stoyanova

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Correspondence to E. B. Saff.

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Dedicated to Stephen J. Gardiner on the occasion of his 60th Birthday.

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The research of the first and fifth authors was supported, in part, by a Bulgarian NSF contract DN02/2-2016. The research of the second author was supported, in part, by a Simons Foundation Grant No. 282207. The research of the third and fourth authors was supported, in part, by the U. S. National Science Foundation under Grant DMS-1516400.

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Boyvalenkov, P.G., Dragnev, P.D., Hardin, D.P. et al. Energy bounds for codes in polynomial metric spaces. Anal.Math.Phys. 9, 781–808 (2019). https://doi.org/10.1007/s13324-019-00313-x

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