Mutually Disjoint 5-Designs From Pless Symmetry Codes


Pless (1972) defines symmetry codes over the field of three elements, called the Pless symmetry codes, and shows that some of these codes yield 5-designs. In this article, we generalize the construction of mutually disjoint Steiner systems studied in Jimbo and Shiromoto (2009) to 5-designs related to a certain class of Pless symmetry codes. As a consequence, we derive new simple 5-designs from the construction.

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  1. Araya, M. 2003. More mutually disjoint Steiner systems S(5, 8, 24). J. Combin. Theory, Ser. A, 102, 201–203.

    MathSciNet  Article  Google Scholar 

  2. Beth, T., C. Charnes, M. Grassl, G. Alber, A. Delgado, and M. Mussinger. 2003. A new class of designs which protect against quantum jumps. Designs Codes Cryptogr., 29, 51–70.

    MathSciNet  Article  Google Scholar 

  3. Colbourn, C. J., and J. H. Dinitz., 2007. Handbook of combinatorial designs, 2nd ed., Boca Raton, FL, CRC Press.

    Google Scholar 

  4. Hall, M. 1967. Combinatorial theory. Waltham, MA, Blaisdell.

    Google Scholar 

  5. Huffman, W., and V. Pless. 2003. Fundamentals of error-correcting codes. Cambridge, Cambridge University Press.

    Google Scholar 

  6. Jimbo, M., and K. Shiromoto. 2009. A construction of mutually disjoint Steiner systems from isomorphic Golay codes. J. Combin. Theory, Ser. A, 116, 1245–1251.

    MathSciNet  Article  Google Scholar 

  7. Kramer, E. S., and S. S. Magliveras. 1974. Some mutually disjoint Steiner systems. J. Combin. Theory, Ser. A, 17, 39–43.

    MathSciNet  Article  Google Scholar 

  8. MacWilliams, F. J., and N. J. A. Sloane. 1978. The theory of error-correcting codes. Amsterdam, North-Holland Publishing Company.

    Google Scholar 

  9. Cannon, J., and W. Bosma (Eds.). 2008. Handbook of Magma functions, Version 2.15. Sydney Australia, University of Sydney.

    Google Scholar 

  10. Pless, V. 1972. Symmetry codes over GF(3) and new five-designs. J. Combin. Theory, Ser. A, 12, 119–142.

    MathSciNet  Article  Google Scholar 

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Correspondence to Keisuke Shiromoto.

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Angata, M., Shiromoto, K. Mutually Disjoint 5-Designs From Pless Symmetry Codes. J Stat Theory Pract 6, 78–87 (2012).

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  • 05B05
  • 94B05


  • Pless symmetry code
  • Self-dual code
  • Simple 5-design