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Energy bounds for codes and designs in Hamming spaces

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Abstract

We obtain universal bounds on the energy of codes and designs in Hamming spaces. Our bounds hold for a large class of potential functions, allow a unified treatment, and can be viewed as a generalization of the Levenshtein bounds for maximal codes.

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Notes

  1. Here the point \(-x\) is the unique point in \(\mathbb {H}(n,2)\) such that \(d(x,-x)=n\).

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Acknowledgments

The authors thank the anonymous referees for valuable remarks. The authors express their gratitude to Erwin Schrödinger International Institute for providing a conducive research atmosphere during their stay when the manuscript was started. The research of P. G. Boyvalenkov was supported, in part, by a Bulgarian NSF Contract I01/0003. The research of P. D. Dragnev was supported, in part, by a Simons Foundation Grant No. 282207. The research of D. P. Hardin and E. B. Saff was supported, in part, by the U. S. National Science Foundation under Grants DMS-1412428 and DMS-1516400. The research of M. M. Stoyanova was supported, in part, by the Science Foundation of Sofia University under Contracts 144/2015 and 57/2016.

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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. Faculty of Mathematics and Natural Sciences, South-Western University, Blagoevgrad, Bulgaria

    P. G. Boyvalenkov

  3. Department of Mathematical Sciences, Indiana-Purdue University, 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

Authors
  1. P. G. Boyvalenkov
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  2. P. D. Dragnev
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  3. D. P. Hardin
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  4. E. B. Saff
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  5. M. M. Stoyanova
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Corresponding author

Correspondence to P. G. Boyvalenkov.

Additional information

This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Coding and Cryptography”.

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Boyvalenkov, P.G., Dragnev, P.D., Hardin, D.P. et al. Energy bounds for codes and designs in Hamming spaces. Des. Codes Cryptogr. 82, 411–433 (2017). https://doi.org/10.1007/s10623-016-0286-4

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  • DOI: https://doi.org/10.1007/s10623-016-0286-4

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