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On Higher Weights and Code Existence

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Cryptography and Coding (IMACC 2009)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 5921))

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Abstract

Several open questions in coding theory relate to non- existence or construction of certain optimal codes. Many previous problems of this kind have been solved by studying possible weight enumerators. A couple of authors in this decade have proposed using higher weights (generalised Hamming weights) to a similar effect. In this paper we suggest one approach based on the weight hierarchy, and it allows us to conduct an extremely rapid computer search to prove that there are exactly two inequivalent [36,8,16] codes. The technique can also be used to gain new information about the weight hierarchy of the putative [72,36,16] code, but not yet enough to say if it exists or not.

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References

  1. Ore, O.: Theory of non-commutative polynomials. Ann. of Math. 34 (1933)

    Google Scholar 

  2. Dodunekov, S.M., Helleseth, T., Manev, N., Ytrehus, Ø.: New bounds on binary linear codes of dimension eight. IEEE Trans. Inform. Theory 33(6), 917–919 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  3. Dougherty, S., Gulliver, A.: Higher weights of self-dual codes. In: Augot, D. (ed.) Workshop on Coding and Cryptography, pp. 177–188 (January 2001)

    Google Scholar 

  4. Dougherty, S., Gulliver, A., Oura, M.: Higher weights and graded rings for binary self-dual codes. Discrete Applied Mathematics 128, 251–261 (2003); Special issue for WCC 2001

    Article  MathSciNet  Google Scholar 

  5. David Forney Jr., G.: Dimension/length profiles and trellis complexity of linear block codes. IEEE Trans. Inform. Theory 40(6), 1741–1752 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  6. Helleseth, T., Kløve, T., Mykkeltveit, J.: The weight distribution of irreducible cyclic codes with block lengths \(n_1((q^l-1)/n)\). Discrete Math. 18, 179–211 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  7. Kløve, T.: Support weight distribution of linear codes. Discrete Math. 106/107, 311–316 (1992)

    Article  Google Scholar 

  8. Luo, Y., Mitrpant, C., Vinck, A.J.H., Chen, K.: Some new characters on the wire-tap channel of type II. IEEE Transactions on Information Theory 51(3), 1222–1229 (2005)

    Article  MathSciNet  Google Scholar 

  9. McKay, B.D.: The nauty page (2002), http://cs.anu.edu.au/people/bdm/nauty/

  10. Ozarow, L.H., Wyner, A.D.: Wire-tap channel II. AT&T Bell Laboratories Technical Journal 63(10), 2135–2157 (1984)

    MATH  Google Scholar 

  11. Schaathun, H.G.: Duality and support weight distributions. IEEE Transactions on Information Theory 50(5), 862–867 (2004)

    Article  MathSciNet  Google Scholar 

  12. Simonis, J.: The effective length of subcodes. Appl. Algebra Engrg. Comm. Comput. 5(6), 371–377 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  13. Wei, V.K.: Generalized Hamming weights for linear codes. IEEE Trans. Inform. Theory 37(5), 1412–1418 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  14. Wei, V.K., Yang, K.: On the generalized Hamming weights of product codes. IEEE Trans. Inform. Theory 39(5), 1709–1713 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  15. Ytrehus, Ø.: Code-buster: A software tool for characterizing abstract codes. Technical report, Dept. of Informatics, University of Bergen (March 1987)

    Google Scholar 

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Authors and Affiliations

  1. Department of Computing, University of Surrey, GU2 7XH, Guildford, Surrey

    Hans Georg Schaathun

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  1. Hans Georg Schaathun
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Editor information

Editors and Affiliations

  1. The Selmer Centre Department of Informatics, University of Bergen, P.O. Box 7800, N-5020, Bergen, Norway

    Matthew G. Parker

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Schaathun, H.G. (2009). On Higher Weights and Code Existence. In: Parker, M.G. (eds) Cryptography and Coding. IMACC 2009. Lecture Notes in Computer Science, vol 5921. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-10868-6_4

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  • DOI: https://doi.org/10.1007/978-3-642-10868-6_4

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-10867-9

  • Online ISBN: 978-3-642-10868-6

  • eBook Packages: Computer ScienceComputer Science (R0)

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