Skip to main content

Unsolved Problems Related to the Covering Radius of Codes

  • Chapter

Abstract

Some of the principal unsolved problems related to the covering radius of codes are described. For example, although it is almost 20 years since it was built, Elwyn Berlekamp’s light-bulb game is still unsolved.

This is a preview of subscription content, log in via an institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. T.A. Brown and J.H. Spencer, “Minimization of ±1 Matrices Under Line Shifts,” Colloq. Math. 23, pp. 165–171 (1971).

    MathSciNet  MATH  Google Scholar 

  2. G.D. Cohen, M.G. Karpovsky, H.F. Mattson, Jr., and J.R. Schatz, “Covering Radius-Survey and Recent Results,” IEEE Trans. Inf. Theory, IT-31, pp. 328–343 (1985).

    Article  MathSciNet  Google Scholar 

  3. G.D. Cohen, A.C. Lobstein, and N.J.A. Sloane, “Further Results on the Covering Radius of Codes,” IEEE Trans. Inf. Theory, to appear.

    Google Scholar 

  4. D.E. Downie and N.J.A Sloane, “The Covering Radius of Cyclic Codes of Length up to 31,” IEEE Trans. Inf. Theory, IT-31, pp. 446–447 (1985).

    Article  MathSciNet  Google Scholar 

  5. Y. Gordon and H.S. Witsenhausen, “On Extensions of the Gale-Berlekamp Switching Problem and Constant of l p Spaces,” Israel J. Math., 11, pp. 216–229 (1972).

    Article  MathSciNet  MATH  Google Scholar 

  6. R.L. Graham and N.J.A. Sloane, “On the Covering Radius of Codes,” IEEE Trans. Inf. Theory, IT-31, pp. 385–401 (1985).

    Article  MathSciNet  Google Scholar 

  7. K.E. Kilby and N.J.A. Sloane, “On the Covering Radius Problem for Codes, I and II,” submitted to SIAM J. Algeb. Discrete Methods.

    Google Scholar 

  8. F.J. MacWilliams and N.J.A. Sloane, The Theory of Error-Correcting Codes, North-Holland, Amsterdam, 2nd printing, 1978.

    Google Scholar 

  9. H.F. Mattson, Jr., “An Improved Upper Bound on Covering Radius,” preprint.

    Google Scholar 

  10. N.J. Patterson and D.H. Wiedemann, “The Covering Radius of the (215, 16) Reed-Muller Code is at Least 16276,” IEEE Trans. Inf Theory, IT-29, pp. 354–356 (1983).

    Article  MathSciNet  Google Scholar 

  11. W.W. Peterson and E.J. Weldon, Jr., Error-Correcting Codes, M.I.T. Press, Cambridge, MA, 2nd ed., 1972.

    MATH  Google Scholar 

  12. O.S. Rothaus, “On Bent Functions,” J. Combinatorial Theory, 20A, pp. 300–305 (1976).

    Article  MathSciNet  Google Scholar 

  13. N.J.A. Sloane, “A New Approach to the Covering Radius of Codes,” J. Combinatorial Theory, to appear.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1987 Springer-Verlag New York Inc.

About this chapter

Cite this chapter

Sloane, N.J.A. (1987). Unsolved Problems Related to the Covering Radius of Codes. In: Cover, T.M., Gopinath, B. (eds) Open Problems in Communication and Computation. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-4808-8_11

Download citation

  • DOI: https://doi.org/10.1007/978-1-4612-4808-8_11

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-9162-6

  • Online ISBN: 978-1-4612-4808-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics