Skip to main navigation menu Skip to main content Skip to site footer

Vol. 52 No. 3 (2026)
Serdica Mathematical Journal

Articles

Locally three-weight property for linear codes and its application

https://doi.org/10.55630/serdica.2026.52.219-248

Published 25-09-2026

  • Shintaro Kawamoto
    • ,
  • Atsuya Kato
    • ,
  • Tatsuya Maruta

Shintaro Kawamoto
Osaka Metropolitan University

Atsuya Kato
Osaka Prefecture University

Tatsuya Maruta
Osaka Metropolitan University

Abstract

An \([n,k,d]_q\) code is a linear code of length \(n\), dimension \(k\) and minimum weight \(d\) over \(\mathbb{F}_q\). A fundamental problem in coding theory is to find \(n_q(k,d)\), the minimum length \(n\) for which an \([n,k,d]_q\) code exists for given \(k,d\) and \(q\). We introduce a new notion ``\(s\)-locally \(w\)-weight\(\pmod{q}\)'' in the usual geometric method for linear codes over \(\mathbb{F}_q\). We give some necessary conditions for \(w=3\) with \(s=1,2\). As an application, we prove the non-existence of some linear codes attaining the Griesmer bound, which determines \(n_8(4,d)\) and \(n_9(4,d)\) for some \(d\).

2020 Mathematics Subject Classification:

94B27, 94B65, 94B05, 51E21

Keywords

optimal linear code, three-weight code, non-existence, geometric method

Full text

Author Details

Shintaro Kawamoto

Department of Mathematics
Osaka Metropolitan University
Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japan
e-mail: ks.hitsujiuma415@gmail.com

Atsuya Kato

Department of Mathematical Sciences
Osaka Prefecture University
Sakai, Osaka 599-8531, Japan
e-mail: atatyktu1031@gmail.com

Tatsuya Maruta

Department of Mathematics
Osaka Metropolitan University
Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japan
e_mail: maruta@omu.ac.jp

References

  1. J. Bierbrauer. Introduction to Coding Theory. Discrete Math. Appl. (Boca Raton) Boca Raton, FL, Chapman & Hall/CRC, 2005.
  2. N. Bono, M. Fujii, T. Maruta. On optimal linear codes of dimension 4, J. Algebra Comb. Discrete Appl. 8, 2 (2021), 73–90.
  3. M. Grassl. Tables of linear codes and quantum codes (electronic table, online), http://www.codetables.de/.
  4. J. H. Griesmer. A bound for error-correcting codes. IBM J. Res. Develop. 4 (1960), 532–542.
  5. 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, 1–3 (1993), 229–268.
  6. R. Hill. A First Course in Coding Theory. Oxford Appl. Math. Comput. Sci. Ser. New York, The Clarendon Press, Oxford University Press, 1986.
  7. R. Hill. Optimal linear codes, In: Cryptography and Coding II, (ed. C. Mitchell) Inst. Math. Appl. Conf. Ser. New Ser., vol. 33. New York, The Clarendon Press, Oxford University Press, 1992, 75–104.
  8. R. Hill. An extension theorem for linear codes. Des. Codes Cryptogr. 17, 1–3 (1999), 151–157.
  9. R. Hill, E. Kolev. A survey of recent results on optimal linear codes. In: Combinatorial Designs and their Applications (eds F. C. Holroyd et al.) Chapman & Hall/CRC Res. Notes Math., vol. 403. Boca Raton, FL, Chapman & Hall/CRC, 1999, 127–152.
  10. R. Hill, P. Lizak. Extensions of linear codes, Proc. IEEE Int. Symposium on Inform. Theory (Whistler, Canada, 1995), 345 pp.
  11. R. Hill, H. Ward. A geometric approach to classifying Griesmer codes. Des. Codes Cryptogr. 44, 1–3 (2007), 169–196.
  12. J. W. P. Hirschfeld. Projective Geometries over Finite Fields. Oxford Math. Monogr. New York, The Clarendon Press, Oxford University Press, 1998.
  13. R. Kanazawa, T. Maruta. On optimal linear codes over F8. Electron. J. Combin. 18, 1 (2011), Paper P34, 27 pp.
  14. H. Kanda, A. Kato, T. Maruta. Locally two-weight property for linear codes and its application, Serdica J. Comput. 17, 2 (2023), 95–106.
  15. K. Kumegawa, T. Okazaki, T. Maruta. On the minimum length of linear codes over the field of 9 elements. Electron. J. Combin. 24, 1 (2017), Paper 1.50, 27 pp.
  16. I. N. Landjev, T. Maruta. On the minimum length of quaternary linear codes of dimension five. Discrete Math., 202, 1–3 (1999), 145–161.
  17. W. Ma, J. Luo. Nonexistence of some four dimensional linear codes attaining the Griesmer bound. Adv. Math. Commun. 18, 2 (2024), 267–282.
  18. T. Maruta. On the nonexistence of q-ary linear codes of dimension five. Des. Codes Cryptogr. 22, 2 (2001), 165–177.
  19. T. Maruta. A new extension theorem for linear codes. Finite Fields Appl. 10, 4 (2004), 674–685.
  20. T. Maruta. Optimal 4-dimensional linear codes over F8. Proceedings of 13th International Workshop on Algebraic and Combinatorial Coding Theory (ACCT 2012), Pomorie, Bulgaria, 2012, 257–262.
  21. T. Maruta. Griesmer bound for linear codes over finite fields, http://mars39.lomo.jp/opu/griesmer.htm.
  22. T. Maruta, A. Kikui, Y. Yoshida. On the uniqueness of (48; 6)-arcs in PG(2; 9). Adv. Math. Commun. 3, 1 (2009), 29–34.
  23. T. Maruta, M. Takeda, K. Kawakami. New sufficient conditions for the extendability of quaternary linear codes. Finite Fields Appl. 14, 3 (2008), 615–634.
  24. G. Solomon, J.J. Stiffler. Algebraically punctured cyclic codes. Information and Control 8 (1965), 170–179.
  25. M. Takenaka, K. Okamoto, T. Maruta. On optimal non-projective ternary linear codes. Discrete Math. 308, 5–6 (2008), 842–854.
  26. H. N. Ward, Divisible codes – a survey, Serdica Math. J. 27, 4 (2001), 263–278.
  27. Y. Yoshida, T. Maruta. An extension theorem for [n; k; d]q codes with gcd(d; q) = 2. Australas. J. Combin. 48 (2010), 117–131.