Journal of the Chungcheong Mathematical Society (충청수학회지)

The Chungcheong Mathematical Society (충청수학회)
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DETERMINATION OF MINIMUM LENGTH OF SOME LINEAR CODES

  • Cheon, Eun Ju (Department of Mathematics and RINS Gyeongsang National University)
  • Received : 2012.10.17
  • Accepted : 2013.01.11
  • Published : 2013.02.15
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Abstract

Hamada ([8]) and Maruta ([17]) proved the minimum length $n_3(6,\;d)=g_3(6,\;d)+1$ for some ternary codes. In this paper we consider such minimum length problem for $q{\geq}4$, and we prove that $n_q(6,\;d)=g_q(6,\;d)+1$ for $d=q^5-q^3-q^2-2q+e$, $1{\leq}e{\leq}q$. Combining this result with Theorem A in [4], we have $n_q(6,\;d)=g-q(6,\;d)+1$ for $q^5-q^3-q^2-2q+1{\leq}d{\leq}q^5-q^3-q^2$ with $q{\geq}4$. Note that $n_q(6,\;d)=g_q(6,\;d)$ for $q^5-q^3-q^2+1{\leq}d{\leq}q^5$ by Theorem 1.2.

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References

  1. A. A. Bruen, Polynomial multiplicities over finite fields and intersection sets, Journal of Combinatorial Theroy, Series A, 60 (1992), 19-33. https://doi.org/10.1016/0097-3165(92)90035-S
  2. E. J. Cheon, T. Kato, S. J. Kim, Nonexistence of [n, 5, d]q codes attaining the Griesmer bound for $q^4-2q^2-2q+1{\leq}d{\leq}q^4-2q^2-q$, Designs, Codes and Cryptography, 36 (2005), 289-299. https://doi.org/10.1007/s10623-004-1720-6
  3. E. J. Cheon, T. Maruta, On the minimum length of some linear codes, Designs, Codes and Cryptography, 43 (2007), 123-135. https://doi.org/10.1007/s10623-007-9070-9
  4. E. J. Cheon, T. Kato, On the minimum length of some linear codes of dimension 6, Bull. Korean Math. Soc., 45 (2008), 419-425. https://doi.org/10.4134/BKMS.2008.45.3.419
  5. E. J. Cheon, The non-existence of Griesmer codes with parameters close to codes of Belov type, Designs, Codes and Cryptography, 61 (2011), 131-139. https://doi.org/10.1007/s10623-010-9443-3
  6. M. Grassl, Bounds on linear codes [n, k, d] over GF(q), available on line: http://www.codetables.de/
  7. N. Hamada., A characterization of some [n, k, d, q]-codes meeting the Griesmer bound using a minihyper in a finite projective geometry, Discrete Math. 116 (1993), 229-268. https://doi.org/10.1016/0012-365X(93)90404-H
  8. N. Hamada., A survey of recent work on characterization of minihypers in PG(t,q) and nonbinary linear codes meeting the Griesmer bound, J. Combin. Inform. & Syst. Sci. 18 (1993), 161-191.
  9. R. Hill, Optimal linear codes, Cryptography and Coding II (ed. C. Mitchell), Oxford Univ. Press, Oxford, 75-104 (1992).
  10. A. Klein, K. Metsch, Parameters for which the Griesmer bound is not sharp, Discrete Math., 307 (2007), 2695-2703. https://doi.org/10.1016/j.disc.2007.01.015
  11. G. Markus, Code Tables: Bounds on the parameters of various types of codes. available on line: http://www.codetables.de/
  12. I. N. Landjev, T. Maruta, On the minimum length of quaternary linear codes of dimension five, Discrete Math., 202 (1999), 145-161. https://doi.org/10.1016/S0012-365X(98)00354-9
  13. Maruta T., A characterization of some minihypers and its application to linear codes, Geometriae Dedicata 74 (1999), 305-311. https://doi.org/10.1023/A:1005076729296
  14. T. Maruta, On the nonexistence of q-ary linear codes of dimension five, Designs, Codes and Cryptography 22 (2001), 165-177. https://doi.org/10.1023/A:1008317022638
  15. T. Maruta, I. N. Landjev, A. Rousseva, On the minimum size of some minihypers and related linear codes, Designs, Codes and Cryptography, 34 (2005), 5-15. https://doi.org/10.1007/s10623-003-4191-2
  16. T. Maruta, Griesmer Bound for Linear Codes over Finite Fields. available on line: http://www.mi.s.osakafu-u.ac.jp/-maruta/griesmer.htm
  17. T. Maruta, Y. Yoshida, Ternary linear codes and quadrics, Electronic Journal of Combinatorics, 16, #R9 (2009).