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ON THE CONSTRUCTION OF OPTIMAL LINEAR CODES OF DIMENSION FOUR

  • Atsuya Kato (Department of Mathematical Sciences Osaka Prefecture University) ;
  • Tatsuya Maruta (Department of Mathematics Osaka Metropolitan University) ;
  • Keita Nomura (Department of Mathematical Sciences Osaka Prefecture University)
  • Received : 2022.09.05
  • Accepted : 2023.04.21
  • Published : 2023.09.30

Abstract

A fundamental problem in coding theory is to find nq(k, d), the minimum length n for which an [n, k, d]q code exists. We show that some q-divisible optimal linear codes of dimension 4 over 𝔽q, which are not of Belov type, can be constructed geometrically using hyperbolic quadrics in PG(3, q). We also construct some new linear codes over 𝔽q with q = 7, 8, which determine n7(4, d) for 31 values of d and n8(4, d) for 40 values of d.

Keywords

Acknowledgement

The authors want to express their gratitude to the referee for the helpful comments and suggestions.

References

  1. S. Ball, Table of bounds on three dimensional linear codes or (n, r)-arcs in PG(2, q), available at https://web.mat.upc.edu/people/simeon.michael.ball/codebounds.html
  2. B. I. Belov, V. N. Logachev, and V. P. Sandimirov, Construction of a class of linear binary codes that attain the Varsamov-Griesmer bound, Problemy Peredaci Informacii 10 (1974), no. 3, 36-44.
  3. J. Bierbrauer, Introduction to Coding Theory, Chapman & Hall, Dordrecht, 2005.
  4. N. Bono, M. Fujii, and T. Maruta, On optimal linear codes of dimension 4, J. Algebra Comb. Discrete Struct. Appl. 8 (2021), no. 2, 73-90. https://doi.org/10.13069/jacodesmath.935947
  5. I. G. Bouyukliev, Y. Kageyama, and T. Maruta, On the minimum length of linear codes over ?5, Discrete Math. 338 (2015), no. 6, 938-953. https://doi.org/10.1016/j.disc.2015.01.010
  6. A. E. Brouwer and M. van Eupen, The correspondence between projective codes and 2-weight codes, Des. Codes Cryptogr. 11 (1997), no. 3, 261-266. https://doi.org/10.1023/A:1008294128110
  7. E. J. Cheon, On the upper bound of the minimum length of 5-dimensional linear codes, Australas. J. Combin. 37 (2007), 225-232.
  8. B. Csajbok and T. Heger, Double blocking sets of size 3q - 1 in PG(2, q), European J. Combin. 78 (2019), 73-89. https://doi.org/10.1016/j.ejc.2019.01.004
  9. S. M. Dodunekov, Optimal linear codes, Doctor Thesis, Sofia, 1985.
  10. M. Grassl, Tables of linear codes and quantum codes, (electronic table, online), http://www.codetables.de/.
  11. J. H. Griesmer, A bound for error-correcting codes, IBM J. Res. Develop. 4 (1960), 532-542. https://doi.org/10.1147/rd.45.0532
  12. R. Hill, Optimal linear codes, in Cryptography and coding, II (Cirencester, 1989), 75-104, Inst. Math. Appl. Conf. Ser. New Ser., 33, Oxford Univ. Press, New York, 1992.
  13. R. Hill and E. Kolev, A survey of recent results on optimal linear codes, in Combinatorial designs and their applications (Milton Keynes, 1997), 127-152, Chapman & Hall/CRC Res. Notes Math., 403, Chapman & Hall/CRC, Boca Raton, FL, 1999.
  14. J. W. P. Hirschfeld, Finite projective spaces of three dimensions, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1985.
  15. W. C. Huffman and V. S. Pless, Fundamentals of Error-Correcting Codes, Cambridge Univ. Press, Cambridge, 2003. https://doi.org/10.1017/CBO9780511807077
  16. Y. Inoue and T. Maruta, Construction of new Griesmer codes of dimension 5, Finite Fields Appl. 55 (2019), 231-237. https://doi.org/10.1016/j.ffa.2018.10.007
  17. Y. Kageyama and T. Maruta, On the geometric constructions of optimal linear codes, Des. Codes Cryptogr. 81 (2016), no. 3, 469-480. https://doi.org/10.1007/s10623-015-0167-2
  18. K. Kumegawa and T. Maruta, Nonexistence of some Griesmer codes over Fq, Discrete Math. 339 (2016), no. 2, 515-521. https://doi.org/10.1016/j.disc.2015.09.030
  19. K. Kumegawa and T. Maruta, Non-existence of some 4-dimensional Griesmer codes over finite fields, J. Algebra Comb. Discrete Struct. Appl. 5 (2018), no. 2, 101-116. https://doi.org/10.13069/jacodesmath.427968
  20. K. Kumegawa, T. Okazaki, and T. Maruta, On the minimum length of linear codes over the field of 9 elements, Electron. J. Combin. 24 (2017), no. 1, Paper No. 1.50, 27 pp. https://doi.org/10.37236/6394
  21. I. N. Landjev and T. Maruta, On the minimum length of quaternary linear codes of dimension five, Discrete Math. 202 (1999), no. 1-3, 145-161. https://doi.org/10.1016/S0012-365X(98)00354-9
  22. T. Maruta, On the minimum length of q-ary linear codes of dimension four, Discrete Math. 208/209 (1999), 427-435. https://doi.org/10.1016/S0012-365X(99)00088-6
  23. T. Maruta, On the nonexistence of q-ary linear codes of dimension five, Des. Codes Cryptogr. 22 (2001), 165-177. https://doi.org/10.1023/A:1008317022638
  24. T. Maruta, Construction of optimal linear codes by geometric puncturing, Serdica J. Comput. 7 (2013), no. 1, 73-80. https://doi.org/10.55630/sjc.2013.7.73-80
  25. T. Maruta, Griesmer bound for linear codes over finite fields, available at http://mars39.lomo.jp/opu/griesmer.htm.
  26. T. Maruta and Y. Oya, On optimal ternary linear codes of dimension 6, Adv. Math. Commun. 5 (2011), no. 3, 505-520. https://doi.org/10.3934/amc.2011.5.505
  27. T. Maruta, M. Shinohara, and M. Takenaka, Constructing linear codes from some orbits of projectivities, Discrete Math. 308 (2008), no. 5-6, 832-841. https://doi.org/10.1016/j.disc.2007.07.045
  28. M. Takenaka, K. Okamoto, and T. Maruta, On optimal non-projective ternary linear codes, Discrete Math. 308 (2008), no. 5-6, 842-854. https://doi.org/10.1016/j.disc.2007.07.044
  29. H. N. Ward, Divisibility of codes meeting the Griesmer bound, J. Combin. Theory Ser. A 83 (1998), no. 1, 79-93. https://doi.org/10.1006/jcta.1997.2864