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On the Nonexistence of q-ary Linear Codes of Dimension Five

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Abstract

There do not exist\([n,5,d]\) codes over the Galois field GF \(\left( q \right)\)attaining the Griesmer bound for \(d = q^4 - 2q^2 - q + 1,2q^4 - 2q^3 - q^2 - q + 1\) for \(q \geqslant 3\) andfor \(d = 3q^4 - 4q^3 - q + 1\) for \(q \geqslant 5\).

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REFERENCES

  1. S.M. Dodunekov, Optimal linear codes, Doctor of Mathematical Sciences Dissertation, Institute of Mathematics, Sofia (1985).

    Google Scholar 

  2. J.H. Griesmer, A bound for error-correcting codes, IBM J. Res. Develop., Vol. 4 (1960) pp. 532–542.

    Google Scholar 

  3. N. Hamada, A characterization of some [n; k; dI q] codes meeting the Griesmer bound using a minihyper in a finite projective geometry, Discrete Math., Vol. 116 (1993) pp. 229–268.

    Google Scholar 

  4. N. Hamada, A survey of recent work on characterization of minihypers in PG(t, q) and nonbinary linear codes meeting the Griesmer bound, J. Comb. Inf. Syst. Sci., Vol. 18 (1993) pp. 161–191.

    Google Scholar 

  5. R. Hill, A First Course in Coding Theory, Oxford University Press, Oxford (1986).

    Google Scholar 

  6. R. Hill, Optimal linear codes, Cryptography and Coding II (C. Mitchell, ed.), Oxford Univ. Press, Oxford (1992) pp. 75–104.

    Google Scholar 

  7. R. Hill and P. Lizak, Extensions of linear codes, Proc. IEEE Int. Syposium on Inform. Theory, Whistler, Canada (1995) p. 345.

  8. J.W.P. Hirschfeld, Projective Geometries over Finite Fields, Clarendon Press,Oxford (1979).

  9. I.N. Landjev, The nonexistence of some optimal ternary codes of dimension five, Designs, Codes and Cryptography, Vol. 15 (1998) pp. 245–258.

    Google Scholar 

  10. I.N. Landjev and T. Maruta, On the minimum length of quaternary linear codes of dimension five, Discrete Math., Vol. 202 (1999) pp. 145–161.

    Google Scholar 

  11. J.H._van Lint, Introduction to Coding Theory, Graduate Texts in Math., Vol. 86, Springer-Verlag, Berlin (1982).

    Google Scholar 

  12. T. Maruta, On the non-existence of linear codes attaining the Griesmer bound, Geom. Dedicata, Vol. 60 (1996) pp. 1–7.

    Google Scholar 

  13. T. Maruta, On the minimum length of q-ary linear codes of dimension five, Geom. Dedicata, Vol. 65 (1997) pp. 299–304.

    Google Scholar 

  14. T. Maruta, On the achievement of the Griesmer bound, Designs, Codes and Cryptography, Vol. 12 (1997) pp. 83–87.

    Google Scholar 

  15. T. Maruta, A characterization of some minihypers and its application to linear codes, Geom. Dedicata, Vol. 74 (1999) pp. 305–311.

    Google Scholar 

  16. T. Maruta, On the minimum length of q-ary linear codes of dimension four, Discrete Math. (to appear).

  17. V.S. Pless and W. Huffman (eds.), Handbook of Coding Theory, North-Holland, Amsterdam (1998).

  18. G. Solomon and J.J. Stiffler, Algebraically punctured cyclic codes, Inform. and Control, Vol. 8 (1965) pp. 170–179.

    Google Scholar 

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  1. Tatsuya Maruta
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Maruta, T. On the Nonexistence of q-ary Linear Codes of Dimension Five. Designs, Codes and Cryptography 22, 165–177 (2001). https://doi.org/10.1023/A:1008317022638

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