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Designs from subcode supports of linear codes

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Abstract

We present new constructions of t-designs by considering subcode supports of linear codes over finite fields. In particular, we prove an Assmus-Mattson type theorem for such subcodes, as well as an automorphism characterization. We derive new t-designs (t  ≤  5) from our constructions.

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References

  1. Assmus E.F. Jr., Mattson H.F. Jr. (1969). New 5-designs. J. Comb. Theory 6, 122–151

    Article  MATH  MathSciNet  Google Scholar 

  2. Britz D., Britz T., Shiromoto K., Sørensen H.K. (2007). The higher weight enumerators of the doubly-even, self-dual [48, 24, 12] code. IEEE Trans. Inform. Theory 53, 2567–2571

    Article  MathSciNet  Google Scholar 

  3. Britz T., Shiromoto K.: Designs from matroids, preprint.

  4. Calderbank R., Delsarte P., Sloane N.J.A. (1991). A strengthening of the Assmus-Mattson theorem. IEEE Trans. Inform. Theory 37, 1261–1268

    Article  MATH  MathSciNet  Google Scholar 

  5. Colbourn C.J., Dinitz J.H. (2007). Handbook of Combinatorial Designs (2nd edn). CRC Press, Boca Raton

    MATH  Google Scholar 

  6. Dowling T.A.: Codes, packings and the critical problem, Applicazioni (Univ. Perugia, Perugia, 1970), Ist. Mat., Univ. Perugia, Perugia (1971) 209–224.

  7. Houghten S., Lam C., Thiel L., Parker R.A. (2003). The extended quadratic residue code is the only (48, 24, 12) self-dual doubly-even code. IEEE Trans. Inform. Theory 49, 53–59

    Article  MATH  MathSciNet  Google Scholar 

  8. Huffman W.C., Pless V.S. (2003). Fundamentals of Error-Correcting Codes. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  9. Kim J., Pless V. (2003). Designs in additive codes over GF(4). Des. Codes Cryptogr. 30, 187–199

    Article  MATH  MathSciNet  Google Scholar 

  10. Kløve T. (1978). The weight distribution of linear codes over GF(q l) having generator matrix over GF(q). Discrete Math. 23, 159–168

    Article  MathSciNet  Google Scholar 

  11. Koch H. (1989). On self-dual, doubly-even codes of length 32. J. Comb. Theory, Ser. A 51, 63–76

    Article  MATH  Google Scholar 

  12. Kramer E.S., Magliveras S.S., Mesner D.M. (1981). t-designs from the large mathieu groups. Discrete Math. 36, 171–189

    MATH  MathSciNet  Google Scholar 

  13. Leon J.S., Pless V.S., Sloane N.J. (1981). On ternary self-dual codes of length 24. IEEE Trans. Inform. Theory 27, 176–180

    Article  MATH  MathSciNet  Google Scholar 

  14. The Magma Computational Algebra System for Algebra, Number Theory and Geometry, Version 2.12, University of Sydney (2005).

  15. MacWilliams F.J., Sloane N.J.: The Theory of Error-Correcting Codes, North-Holland Mathematical Library, 16., North-Holland Publishing Company, Amsterdam (1978).

  16. Milenkovic O., Coffey S.T., Compton K.J. (2003). The third support weight enumerators of the doubly-even, self-dual [32, 16, 8] codes. IEEE Trans. Inform. Theory 49: 741–746

    Article  MathSciNet  Google Scholar 

  17. Pless V.S., Huffman W.C. (eds). (1998). Handbook of Coding Theory II. Elsevier Science B.V., Amsterdam

    Google Scholar 

  18. Shiromoto K. (1996). A new MacWilliams type identity for linear codes. Hokkaido Math. J. 25, 651–656

    MATH  MathSciNet  Google Scholar 

  19. Shiromoto K. (1999). The weight enumerator of linear codes over GF(q m) having generator matrix over GF(q). Des. Codes Cryptogr. 16, 87–92

    Article  MATH  MathSciNet  Google Scholar 

  20. Tsfasman M.A., Vlădut S.G. (1995). Geometric approach to higher weights. IEEE Trans. Inform. Theory 41, 1564–1588

    Article  MATH  MathSciNet  Google Scholar 

  21. van Lint J.H., Wilson R.M. (1992). A Course in Combinatorics. Cambridge University Press, Cambridge

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics, University of New South Wales, Sydney, NSW, 2052, Australia

    Thomas Britz

  2. Department of Information Systems, Aichi Prefectural University, Nagakute, Aichi, 480-1198, Japan

    Keisuke Shiromoto

Authors
  1. Thomas Britz
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  2. Keisuke Shiromoto
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Correspondence to Keisuke Shiromoto.

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Communicated by Q. Xiang.

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Britz, T., Shiromoto, K. Designs from subcode supports of linear codes. Des. Codes Cryptogr. 46, 175–189 (2008). https://doi.org/10.1007/s10623-007-9145-7

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  • DOI: https://doi.org/10.1007/s10623-007-9145-7

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