Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
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EXTREMAL TYPE I ADDITIVE SELF-DUAL CODES OVER GF(4) WITH NEAR-MINIMAL SHADOW

  • Received : 2018.08.31
  • Accepted : 2018.12.12
  • Published : 2018.12.30
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Abstract

In this paper, we define near-minimal shadow and study the existence problem of extremal Type I additive self-dual codes over GF(4) with near-minimal shadow. We prove that there is no such codes if the code length n = 6m+1($m{\geq}0$), $n=6m+5(m{\geq}1)$.

Keywords

TABLE 1. Non-existence of extremal(or near-extremal) Type I additive self-dual codes over GF (4) with minimal(or near-minimal) shadow of length n = 6m+ p

E1KKMK_2018_v26n4_729_t0001.png 이미지

References

  1. E.P. Bautista, P. Gaborit, J.-L. Kim, J.L. Walker, s-extremal additive codes, Adv. Math. Commun. 1 (2007), 111-130. https://doi.org/10.3934/amc.2007.1.111
  2. W. Bosma, J. Cannon, C. Playoust, The Magma algebra system I: The user language, J. Symbolic Comput. 24 (1997), 235-265. https://doi.org/10.1006/jsco.1996.0125
  3. P. Gaborit, W.C. Huffman, J.-L. Kim, V. Pless, On additive GF(4) codes, in: A. Barg, S. Litsyn (Eds.), Codes and Association Schemes, DIMACS Series in Discrete Mathematics and Theoretical Computer Science 56, American Mathematical Society, Providence, RI, 2001, 135-149.
  4. S. Han, Additive self-dual codes over GF(4) with minimal shadow, MDPI Information 9 81 (2018), 1-11. https://doi.org/10.3390/info9010001
  5. S. Han, Near-Extremal Type I Self-Dual Codes with Minimal Shadow over GF(2) and GF(4), MDPI Information 9 172 (2018), 1-12. https://doi.org/10.3390/info9010001
  6. S. Han, On the extremal Type I binary self-dual codes with near-minimal shadow, submitted.
  7. G. Hhn, Self-dual codes over the Kleinian four group, Math. Ann. 327 (2003), 227-255. https://doi.org/10.1007/s00208-003-0440-y
  8. W.C. Huffman, On the classification and enumeration of self-dual codes, Finite Fields Appl. 11 (2005), 451-490. https://doi.org/10.1016/j.ffa.2005.05.012
  9. E.M. Rains, Shadow bounds for self-dual codes, IEEE Trans. Inform. Theory 44 (1998), 134-139. https://doi.org/10.1109/18.651000