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The Nonexistence of Ternary [284, 6, 188] Codes

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Abstract

Let [n, k, d] q codes be linear codes of length n, dimension k, and minimum Hamming distance d over GF(q). Let n q (k, d) be the smallest value of n for which there exists an [n, k, d] q code. It is known from [1, 2] that 284 ≤ n 3(6, 188) ≤ 285 and 285 ≤ n 3(6, 189) ≤ 286. In this paper, the nonexistence of [284, 6, 188]3 codes is proved, whence we get n 3(6, 188) = 285 and n 3(6, 189) = 286.

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REFERENCES

  1. Hamada, N. and Helleseth, T., The Nonexistence of Some Ternary Linear Codes and Update of the Bounds for n3(6; d), 1 ≤ d ≤ 243, Math. Japon., 2000, vol. 52, no. 1, pp. 31–43.

    Google Scholar 

  2. Maruta, T., Table on n3(6; d) [electronic]. Available online at http://www.geocities.com/mars39.geo/griesmer.htm.

  3. Hill, R. and Newton, D.E., Optimal Ternary Linear Codes, Designs Codes Crypt., 1992, vol. 2, pp. 137–157.

    Google Scholar 

  4. van Eupen, M., Five New Optimal Ternary Linear Codes, IEEE Trans. Inf. Theory, 1994, vol. 40, no. 1, pp. 193.

    Google Scholar 

  5. van Eupen, M., Some New Results for Ternary Linear Codes of Dimension 5 and 6, IEEE Trans. Inf. Theory, 1995, vol. 41, no. 6, pp. 2048–2051.

    Google Scholar 

  6. van Eupen, M. and Hill, R., An Optimal Ternary [69; 5; 45] Code and Related Codes, Designs Codes Crypt., 1994, vol. 4, pp. 271–282.

    Google Scholar 

  7. Bogdanova, G. and Boukliev, I., New Linear Codes of Dimension 5 over GF(3), in Proc. 4th Int. Workshop on Algebraic and Combinatorial Coding Theory, Novgorod, Russia, 1994, pp. 41–43.

  8. van Eupen, M., Hamada, N., and Watamori, Y., The Nonexistence of Ternary [50; 5; 32] Codes, Designs Codes Crypt., 1996, vol. 7, pp. 235–237.

  9. Hamada, N. and Watamori, Y., The Nonexistence of [71; 5; 46]3 Codes, J. Stat. Plan. Inference, 1996, vol. 52, pp. 379–394.

    Google Scholar 

  10. Landgev, I., The Nonexistence of Some Optimal Ternary Codes of Dimension Five, Designs Codes Crypt., 1998, vol. 15, pp. 245–258.

    Google Scholar 

  11. Hamada, N., A Survey of Recent Work on Characterization of Minihypers in PG(t; q) andNonbinary Linear Codes Meeting the Griesmer Bound, J.Comb. Inf. Syst.Sci., 1993, vol. 18, pp. 229–268.

    Google Scholar 

  12. Daskalov, R.N., Bounds on the Minimum Length for Ternary Linear Codes of Dimension Six, in Mathematics and Education in Mathematics, Sofia: BAN, 1993, pp. 15–22.

  13. Hamada, N. and Watamori, Y., The Nonexistence of Some Ternary Linear Codes of Dimension 6 and the Bounds for n3(6; d), 1 ≤ d ≤ 243, Math. Japon., 1996, vol. 43, no. 3, pp. 577–593.

    Google Scholar 

  14. Boukliev, I., Some New Optimal Ternary Linear Codes, Designs Codes Crypt., 1997, vol. 12, pp. 5–11.

    Google Scholar 

  15. Brouwer, A.E. and van Eupen, M., The Correspondence between Projective Codes and 2-Weight Codes, Designs Codes Crypt., 1997, vol. 11, pp. 262–266.

    Google Scholar 

  16. Daskalov, R.N., The Non-existence of Ternary Linear [158; 6; 104] and [203; 6; 134] Codes, in Proc. 5th Int. Workshop on Algebraic and Combinatorial Coding Theory, Sozopol, Bulgaria, 1996, pp. 111–116.

  17. Hamada, N., The Nonexistence of Ternary [29; 6; 17] Codes, Math. Japon., 1997, vol. 46, pp. 253–264.

    Google Scholar 

  18. Hamada, N. and van Eupen, M., The Nonexistence of Ternary [38; 6; 23] Codes, Designs Codes Crypt., 1998, vol. 13, pp. 165–172.

    Google Scholar 

  19. Hill, R. and Jones, C., The Nonexistence of Ternary [47; 6; 29] Codes, in Proc. 2nd Int. Workshop OC'98, Sozopol, Bulgaria, 1998, pp. 90–97.

  20. Hamada, N. and Watamori, Y., The Nonexistence of Ternary [79; 6; 51] Codes, J. Stat. Plan. Inference, 1998, vol. 72, pp. 323–332.

    Google Scholar 

  21. Hamada, N. and Helleseth, T., The Nonexistence of Ternary [97; 6; 63] Codes, J. Stat. Plan. Inference, 2002, vol. 106, pp. 485–507.

    Google Scholar 

  22. Hamada, N., Helleseth, T., Martinsen, H., and Ytrehus, Ø., There Is No Ternary [28; 6; 16] Code, IEEE Trans. Inf. Theory, 2000, vol. 46, no. 5, pp. 1550–1554.

    Google Scholar 

  23. Jones, C., Optimal Ternary Linear Codes, PhD Thesis, Greater Manchester: University of Salford, 2000.

    Google Scholar 

  24. Maruta, T., The Nonexistence of Ternary Linear Codes of Dimension 6, preprint.

  25. Griesmer, J.H., A Bound for Error-Correcting Codes, IBM J. Res. Develop., 1960, vol. 4, pp. 532–542.

    Google Scholar 

  26. Solomon, G. and Stiffer, J.J., Algebraically Punctured Cyclic Codes, Inf. Control, 1965, vol. 8, no. 2, pp. 170–179.

    Google Scholar 

  27. MacWilliams, F.J. and Sloane, N.J.A., The Theory of Error-Correcting Codes, Amsterdam: North-Holland, 1977. Translated under the title Teoriya kodov, ispravlyayushchikh oshibki, Moscow: Svyaz', 1979.

    Google Scholar 

  28. Dodunekov, S.M., Minimum Block Length of a Linear q-ary Code with Specified Dimension and Code Distance, Probl. Peredachi Inf., 1984, vol. 20, no. 4, pp. 11–22 [Probl. Inf. Trans. (Engl. Transl.), 1984, vol. 20, no. 4, pp. 239–249].

    Google Scholar 

  29. van Eupen, M. and Lisonek, P., Classification of Some Optimal Ternary Linear Codes of Small Length, Designs Codes Crypt., 1997, vol. 10, pp. 63–84.

    Google Scholar 

  30. Lidl, R. and Niederreiter, H., Finite Fields, Encyclopedia of Mathematics and Its Applications, vol. 20, Reading: Addison-Wesley, 1983.

    Google Scholar 

  31. Brouwer, A.E., Linear Code Bounds [electronic table]. Available online at http://www.win.tue.nl/~aeb/voorlincod.html.

  32. Hamada, N. and Helleseth, T., The Uniqueness of [87; 5; 57; 3] Codes and the Nonexistence of [258; 6; 171; 3] Codes, J. Stat. Plan. Inference, 1996, vol. 56, pp. 105–127.

    Google Scholar 

  33. van Eupen, M., Four Non-existence Results for Ternary Linear Codes, IEEE Trans. Inf. Theory, 1995, vol. 41, no. 3, pp. 800–805.

    Google Scholar 

  34. Hill, R. and Newton, D.E., Some Optimal Ternary Linear Codes, Ars Comb., Ser. A, 1988, vol. 25, pp. 61–72.

    Google Scholar 

  35. Bouyukliev, I. and Simonis, J., Some New Results for Optimal Ternary Linear Codes, IEEE Trans. Inf. Theory, 2002, vol. 48, no. 4, pp. 981–985.

    Google Scholar 

  36. Ward, H.N., Divisibility of Codes Meeting the Griesmer Bound, J. Comb. Theory, Ser. A, 1998, vol. 83, no. 1, pp. 79–93.

    Google Scholar 

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  1. Department of Mathematics, Technical University of Gabrovo, Bulgaria

    R. Daskalov & E. Metodieva

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  1. R. Daskalov
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Daskalov, R., Metodieva, E. The Nonexistence of Ternary [284, 6, 188] Codes. Problems of Information Transmission 40, 135–146 (2004). https://doi.org/10.1023/B:PRIT.0000043927.19508.8b

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