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Energy bounds for codes and designs in Hamming spaces

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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Correspondence to P. G. Boyvalenkov.

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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