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Energy bounds for codes and designs in Hamming spaces

Designs, Codes and Cryptography volume 82, pages 411–433 (2017)Cite this article

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

Keywords

  • Hamming space
  • Potential functions
  • Energy of a code
  • Error-correcting codes
  • \(\tau \)-designs

Mathematics Subject Classification

  • 74G65
  • 94B65
  • 52A40
  • 05B30