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On optimal linear codes of dimension 4

Yıl 2021, Cilt: 8 Sayı: 2, 73 - 90, 20.05.2021
https://doi.org/10.13069/jacodesmath.935947

Öz

In coding theory, the problem of finding the shortest linear codes for a fixed set of parameters is central. Given the dimension $k$, the minimum weight $d$, and the order $q$ of the finite field $\bF_q$ over which the code is defined, the function $n_q(k, d)$ specifies the smallest length $n$ for which an $[n, k, d]_q$ code exists. The problem of determining the values of this function is known as the problem of optimal linear codes. Using the geometric methods through projective geometry, we determine $n_q(4,d)$ for some values of $d$ by constructing new codes and by proving the nonexistence of linear codes with certain parameters.

Kaynakça

  • [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] A. Betten, E. J. Cheon, S. J. Kim, T. Maruta, The classification of (42; 6)8 arcs, Adv. Math. Commun. 5 (2011) 209–223.
  • [3] I. Bouyukliev, Y. Kageyama, T. Maruta, On the minimum length of linear codes over F5, Discrete Math. 338 (2015) 938–953.
  • [4] A. E. Brouwer, M. van Eupen, The correspondence between projective codes and 2-weight codes, Des. Codes Cryptogr. 11 (1997) 261–266.
  • [5] M. van Eupen, R. Hill, An optimal ternary [69; 5; 45]3 codes and related codes, Des. Codes Cryptogr. 4 (1994) 271–282.
  • [6] M. Fujii, Nonexistence of some Griesmer codes of dimension 4, Master Thesis, Osaka Prefecture University (2019).
  • [7] M. Grassl, "Bounds on the minimum distance of linear codes and quantum codes." Online available at http://www.codetables.de.
  • [8] J. H. Griesmer, A bound for error-correcting codes, IBM J. Res. Develop. 4 (1960) 532–542.
  • [9] 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.
  • [10] R. Hill, Optimal linear codes, In: C. Mitchell(Ed.), Cryptography and Coding II, Oxford Univ. Press, Oxford (1992) 75–104.
  • [11] R. Hill, E. Kolev, A survey of recent results on optimal linear codes, In: Combinatorial Designs and their Applications, F.C. Holroyd et al. Ed., Chapman and Hall/CRC Press Research Notes in Mathematics, CRC Press. Boca Raton (1999) 127–152.
  • [12] J. W. P. Hirschfeld, Finite Projective Spaces of Three Dimensions, Clarendon Press, Oxford (1985).
  • [13] J. W. P. Hirschfeld, Projective Geometries over Finite Fields, Clarendon Press, Oxford, second edition (1998).
  • [14] C. Jones, A. Matney, H. Ward, Optimal four-dimensional codes over GF(8), Electron. J. Combin. 13 (2006) #R43, .
  • [15] Y. Kageyama, T. Maruta, On the geometric constructions of optimal linear codes, Des. Codes Cryptogr., 81 (2016) 469–480.
  • [16] R. Kanazawa, T. Maruta, On optimal linear codes over F8, Electronic J. Combin., 18(1) (2011) #P34
  • [17] 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) #P1.50.
  • [18] K. Kumegawa, T. Maruta, Nonexistence of some Griesmer codes over Fq, Discrete Math. 339 (2016) 515–521.
  • [19] K. Kumegawa, T. Maruta, Non-existence of some 4-dimensional Griesmer codes over finite fields, J. Algebra Comb. Discrete Struct. Appl. 5 (2018) 101–116.
  • [20] I. Landjev, L. Storme, A study of (x(q + 1); x; 2; q)-minihypers, Des. Codes Cryptogr. 54 (2010) 135–147.
  • [21] T. Maruta, On the minimum length of q-ary linear codes of dimension four, Discrete Math., 208/209 (1999) 427–435.
  • [22] T. Maruta, On the nonexistence of q-ary linear codes of dimension five, Des. Codes Cryptogr. 22 (2001) 165–177.
  • [23] T. Maruta, A new extension theorem for linear codes, Finite Fields Appl. 10 (2004) 674–685.
  • [24] 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.
  • [25] T. Maruta, Construction of optimal linear codes by geometric puncturing, Serdica J. Computing 7 (2013) 73–80.
  • [26] T. Maruta, Griesmer bound for linear codes over finite fields, available at http://mars39.lomo.jp/opu/griesmer.htm.
  • [27] T. Maruta, Y. Oya, On optimal ternary linear codes of dimension 6, Adv. Math. Commun. 5 (2011) 505–520.
  • [28] T. Maruta, M. Shinohara, M. Takenaka, Constructing linear codes from some orbits of projectivities,Discrete Math. 308 (2008) 832–841.
  • [29] T. Maruta, T. Tanaka, H. Kanda, Some generalizations of extension theorems for linear codes over finite fields, Australas. J. Combin. 60 (2014) 150–157.
  • [30] T. Maruta, Y. Yoshida, A generalized extension theorem for linear codes, Des. Codes Cryptogr. 62 (2012) 121–130.
  • [31] M. Takenaka, K. Okamoto, T. Maruta, On optimal non-projective ternary linear codes, Discrete Math. 308 (2008) 842–854.
  • [32] 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.
Yıl 2021, Cilt: 8 Sayı: 2, 73 - 90, 20.05.2021
https://doi.org/10.13069/jacodesmath.935947

Öz

Kaynakça

  • [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] A. Betten, E. J. Cheon, S. J. Kim, T. Maruta, The classification of (42; 6)8 arcs, Adv. Math. Commun. 5 (2011) 209–223.
  • [3] I. Bouyukliev, Y. Kageyama, T. Maruta, On the minimum length of linear codes over F5, Discrete Math. 338 (2015) 938–953.
  • [4] A. E. Brouwer, M. van Eupen, The correspondence between projective codes and 2-weight codes, Des. Codes Cryptogr. 11 (1997) 261–266.
  • [5] M. van Eupen, R. Hill, An optimal ternary [69; 5; 45]3 codes and related codes, Des. Codes Cryptogr. 4 (1994) 271–282.
  • [6] M. Fujii, Nonexistence of some Griesmer codes of dimension 4, Master Thesis, Osaka Prefecture University (2019).
  • [7] M. Grassl, "Bounds on the minimum distance of linear codes and quantum codes." Online available at http://www.codetables.de.
  • [8] J. H. Griesmer, A bound for error-correcting codes, IBM J. Res. Develop. 4 (1960) 532–542.
  • [9] 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.
  • [10] R. Hill, Optimal linear codes, In: C. Mitchell(Ed.), Cryptography and Coding II, Oxford Univ. Press, Oxford (1992) 75–104.
  • [11] R. Hill, E. Kolev, A survey of recent results on optimal linear codes, In: Combinatorial Designs and their Applications, F.C. Holroyd et al. Ed., Chapman and Hall/CRC Press Research Notes in Mathematics, CRC Press. Boca Raton (1999) 127–152.
  • [12] J. W. P. Hirschfeld, Finite Projective Spaces of Three Dimensions, Clarendon Press, Oxford (1985).
  • [13] J. W. P. Hirschfeld, Projective Geometries over Finite Fields, Clarendon Press, Oxford, second edition (1998).
  • [14] C. Jones, A. Matney, H. Ward, Optimal four-dimensional codes over GF(8), Electron. J. Combin. 13 (2006) #R43, .
  • [15] Y. Kageyama, T. Maruta, On the geometric constructions of optimal linear codes, Des. Codes Cryptogr., 81 (2016) 469–480.
  • [16] R. Kanazawa, T. Maruta, On optimal linear codes over F8, Electronic J. Combin., 18(1) (2011) #P34
  • [17] 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) #P1.50.
  • [18] K. Kumegawa, T. Maruta, Nonexistence of some Griesmer codes over Fq, Discrete Math. 339 (2016) 515–521.
  • [19] K. Kumegawa, T. Maruta, Non-existence of some 4-dimensional Griesmer codes over finite fields, J. Algebra Comb. Discrete Struct. Appl. 5 (2018) 101–116.
  • [20] I. Landjev, L. Storme, A study of (x(q + 1); x; 2; q)-minihypers, Des. Codes Cryptogr. 54 (2010) 135–147.
  • [21] T. Maruta, On the minimum length of q-ary linear codes of dimension four, Discrete Math., 208/209 (1999) 427–435.
  • [22] T. Maruta, On the nonexistence of q-ary linear codes of dimension five, Des. Codes Cryptogr. 22 (2001) 165–177.
  • [23] T. Maruta, A new extension theorem for linear codes, Finite Fields Appl. 10 (2004) 674–685.
  • [24] 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.
  • [25] T. Maruta, Construction of optimal linear codes by geometric puncturing, Serdica J. Computing 7 (2013) 73–80.
  • [26] T. Maruta, Griesmer bound for linear codes over finite fields, available at http://mars39.lomo.jp/opu/griesmer.htm.
  • [27] T. Maruta, Y. Oya, On optimal ternary linear codes of dimension 6, Adv. Math. Commun. 5 (2011) 505–520.
  • [28] T. Maruta, M. Shinohara, M. Takenaka, Constructing linear codes from some orbits of projectivities,Discrete Math. 308 (2008) 832–841.
  • [29] T. Maruta, T. Tanaka, H. Kanda, Some generalizations of extension theorems for linear codes over finite fields, Australas. J. Combin. 60 (2014) 150–157.
  • [30] T. Maruta, Y. Yoshida, A generalized extension theorem for linear codes, Des. Codes Cryptogr. 62 (2012) 121–130.
  • [31] M. Takenaka, K. Okamoto, T. Maruta, On optimal non-projective ternary linear codes, Discrete Math. 308 (2008) 842–854.
  • [32] 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.
Toplam 32 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Mühendislik
Bölüm Makaleler
Yazarlar

Nanami Bono Bu kişi benim

Maya Fujii Bu kişi benim

Tatsuya Maruta Bu kişi benim 0000-0001-7858-0787

Yayımlanma Tarihi 20 Mayıs 2021
Yayımlandığı Sayı Yıl 2021 Cilt: 8 Sayı: 2

Kaynak Göster

APA Bono, N., Fujii, M., & Maruta, T. (2021). On optimal linear codes of dimension 4. Journal of Algebra Combinatorics Discrete Structures and Applications, 8(2), 73-90. https://doi.org/10.13069/jacodesmath.935947
AMA Bono N, Fujii M, Maruta T. On optimal linear codes of dimension 4. Journal of Algebra Combinatorics Discrete Structures and Applications. Mayıs 2021;8(2):73-90. doi:10.13069/jacodesmath.935947
Chicago Bono, Nanami, Maya Fujii, ve Tatsuya Maruta. “On Optimal Linear Codes of Dimension 4”. Journal of Algebra Combinatorics Discrete Structures and Applications 8, sy. 2 (Mayıs 2021): 73-90. https://doi.org/10.13069/jacodesmath.935947.
EndNote Bono N, Fujii M, Maruta T (01 Mayıs 2021) On optimal linear codes of dimension 4. Journal of Algebra Combinatorics Discrete Structures and Applications 8 2 73–90.
IEEE N. Bono, M. Fujii, ve T. Maruta, “On optimal linear codes of dimension 4”, Journal of Algebra Combinatorics Discrete Structures and Applications, c. 8, sy. 2, ss. 73–90, 2021, doi: 10.13069/jacodesmath.935947.
ISNAD Bono, Nanami vd. “On Optimal Linear Codes of Dimension 4”. Journal of Algebra Combinatorics Discrete Structures and Applications 8/2 (Mayıs 2021), 73-90. https://doi.org/10.13069/jacodesmath.935947.
JAMA Bono N, Fujii M, Maruta T. On optimal linear codes of dimension 4. Journal of Algebra Combinatorics Discrete Structures and Applications. 2021;8:73–90.
MLA Bono, Nanami vd. “On Optimal Linear Codes of Dimension 4”. Journal of Algebra Combinatorics Discrete Structures and Applications, c. 8, sy. 2, 2021, ss. 73-90, doi:10.13069/jacodesmath.935947.
Vancouver Bono N, Fujii M, Maruta T. On optimal linear codes of dimension 4. Journal of Algebra Combinatorics Discrete Structures and Applications. 2021;8(2):73-90.