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Pivoting and Extension pp 133–147Cite as

Derivation of a bound for error-correcting codes using pivoting techniques

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Part of the book series: Mathematical Programming Studies ((MATHPROGRAMM,volume 1))

Abstract

Using an integer programming formulation of the problem of finding a binary linear error-correcting code of minimum length n with 2k code words and minimum distance d, a lower bound on n is constructed. The bound is shown to be equivalent to one previously obtained. Possible extensions using this approach are indicated.

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References

  1. L.D. Baumert and R.J. McEliece, “A note on the Griesmer bound”, IEEE Transactions on Information Theory 19 (1973) 134–135.

    Article  MATH  MathSciNet  Google Scholar 

  2. A.B. Fontaine and W.W. Peterson, “Group code equivalence and optimum codes”, IRE Transactions on Information Theory 5 (1959) Special Supplement, pp. 60–70.

    Article  Google Scholar 

  3. R.E. Gomory, “An algorithm for integer solutions to linear programs,” in Recent advances in mathematical programming, Eds. R.L. Graves and P. Wolfe (McGraw-Hill, New York, 1963) pp. 269–302.

    Google Scholar 

  4. R.E. Gomory, “Faces of an integer polyhedron”, Proceedings of the National Academy of Sciences 57 (1967) 16–18.

    Article  MATH  MathSciNet  Google Scholar 

  5. R.E. Gomory, “Some polyhedra related to combinatorial problems”, Linear Algebra and its Applications 2 (1969) 451–558.

    Article  MATH  MathSciNet  Google Scholar 

  6. R.E. Gomory, “Properties of a class of integer polyhedra”, in Integer and non-linear programming, Ed. J. Abadie (North-Holland, Amsterdam, 1970) pp. 353–365

    Google Scholar 

  7. R.E. Gomory and E.L. Johnson, “Some continuous functions related to corner polyhedra, I”, Mathematical Programming 3 (1972) 23–86.

    Article  MATH  MathSciNet  Google Scholar 

  8. R.E. Gomory and E.L. Johnson, “Some continuous functions related to corner polyhedra, II”, Mathematical Programming 3 (1972) 359–389.

    Article  MATH  MathSciNet  Google Scholar 

  9. J.H. Griesmer, “A bound for error-correcting codes”, IBM Journal of Research and Development 4 (1960) 532–542.

    Article  MATH  MathSciNet  Google Scholar 

  10. H.J. Helgert and R.D. Stinoff, “Minimum-distance bounds for binary linear codes”, IEEE Transactions on Information Theory 19 (1973) 344–356.

    Article  MATH  Google Scholar 

  11. W.W. Peterson and E.J. Weldon, Jr., Error-correcting codes, 2nd ed. (The MIT Press, Cambridge, Mass., 1972).

    MATH  Google Scholar 

  12. D. Slepian, “A class of binary signalling alphabets”, Bell System Technical Journal 35 (1956) 203–234.

    MathSciNet  Google Scholar 

  13. G. Solomon, “Algebraic coding theory”, in: Communication Theory, Ed. A.V. Balakrishnan (McGraw-Hill, New York, 1968) ch. 5, pp. 216–271.

    Google Scholar 

  14. G. Solomon and J. Stiffler, “Algebraically punctured codes”, Information and Control 8 (1965) 170–179.

    Article  MATH  MathSciNet  Google Scholar 

  15. A.W. Tucker, “A combinatorial equivalence of matrices”, in: Combinatorial analysis, Proceedings of Symposia in Applied Mathematics X, Eds R. Bellman and M. Hall (Am. Math. Soc., Providence, R.I., 1960) pp. 129–140.

    Google Scholar 

  16. A.W. Tucker, “Combinatorial theory underlying linear programs”, in Recent advances in mathematical programming, Eds. R.L. Graves and P. Wolfe (McGraw-Hill, New York, 1963) pp. 1–16.

    Google Scholar 

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Authors and Affiliations

  1. IBM Thomas J. Watson Research Center, Yorktown Heights, New York, USA

    James H. Griesmer

Authors
  1. James H. Griesmer
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M. L. Balinski

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© 1974 The Mathematical Programming Society

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Griesmer, J.H. (1974). Derivation of a bound for error-correcting codes using pivoting techniques. In: Balinski, M.L. (eds) Pivoting and Extension. Mathematical Programming Studies, vol 1. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0121245

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  • DOI: https://doi.org/10.1007/BFb0121245

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-00756-9

  • Online ISBN: 978-3-642-00758-3

  • eBook Packages: Springer Book Archive

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