On optimal linear codes of dimension 4
DOI:
https://doi.org/10.13069/jacodesmath.935947Keywords:
Optimal linear codes,Griesmer bound,Geometric methodAbstract
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 $\mathbb{F}_q$ 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.
Received: 23 April 2020 | Accepted: 24 November 2020Downloads
References
S. Ball, Table of bounds on three dimensional linear codes or $(n,r)$-arcs in $\mathrm{PG}(2,q)$, online resource, https://web.mat.upc.edu/people/simeon.michael.ball/codebounds.html.
A. Betten, E. J. Cheon, S. J. Kim, and T. Maruta, The classification of $(42,6)_8$ arcs, Adv. Math. Commun. 5(2) (2011) 209–223.
I. Bouyukliev, Y. Kageyama, and T. Maruta, On the minimum length of linear codes over $\mathbb{F}_5$, Discrete Math. 338(6) (2015) 938–953.
A. E. Brouwer and M. van Eupen, The correspondence between projective codes and 2-weight codes, Des. Codes Cryptogr. 11(3) (1997) 261–266.
M. van Eupen and R. Hill, An optimal ternary $[69,5,45]_3$ code and related codes, Des. Codes Cryptogr. 4(3) (1994) 271–282.
M. Fujii, Nonexistence of some Griesmer codes of dimension 4, Master's Thesis, Osaka Prefecture University (2019).
M. Grassl, Bounds on the minimum distance of linear codes and quantum codes, online database, http://www.codetables.de.
J. H. Griesmer, A bound for error-correcting codes, IBM J. Res. Develop. 4(5) (1960) 532–542.
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.
R. Hill, Optimal linear codes, in: C. Mitchell (Ed.), Cryptography and Coding II, Oxford University Press, Oxford (1992) 75–104.
R. Hill and E. Kolev, A survey of recent results on optimal linear codes, in: Combinatorial Designs and Their Applications, Chapman & Hall/CRC Res. Notes Math., Vol. 403, CRC Press, Boca Raton, FL (1999) 127–152.
J. W. P. Hirschfeld, Finite Projective Spaces of Three Dimensions, Oxford University Press, Oxford (1985).
J. W. P. Hirschfeld, Projective Geometries over Finite Fields, 2nd ed., Oxford University Press, Oxford (1998).
C. Jones, A. Matney, and H. Ward, Optimal four-dimensional codes over $\mathrm{GF}(8)$, Electron. J. Combin. 13(1) (2006) Research Paper 43.
Y. Kageyama and T. Maruta, On the geometric constructions of optimal linear codes, Des. Codes Cryptogr. 81(3) (2016) 469–480.
R. Kanazawa and T. Maruta, On optimal linear codes over $\mathbb{F}_8$, Electron. J. Combin. 18(1) (2011) Paper 34.
K. Kumegawa, T. Okazaki, and T. Maruta, On the minimum length of linear codes over the field of 9 elements, Electron. J. Combin. 24(1) (2017) Paper 1.50.
K. Kumegawa and T. Maruta, Nonexistence of some Griesmer codes over $\mathbb{F}_q$, Discrete Math. 339(1) (2016) 515–521.
K. Kumegawa and T. Maruta, Non-existence of some 4-dimensional Griesmer codes over finite fields, J. Algebra Comb. Discrete Struct. Appl. 5(2) (2018) 101–116.
I. Landjev and L. Storme, A study of $(x(q+1),x;2,q)$-minihypers, Des. Codes Cryptogr. 54(2) (2010) 135–147.
T. Maruta, On the minimum length of $q$-ary linear codes of dimension four, Discrete Math. 208-209 (1999) 427–435.
T. Maruta, On the nonexistence of $q$-ary linear codes of dimension five, Des. Codes Cryptogr. 22(2) (2001) 165–177.
T. Maruta, A new extension theorem for linear codes, Finite Fields Appl. 10(4) (2004) 674–685.
T. Maruta, Optimal 4-dimensional linear codes over $\mathbb{F}_8$, in: Proceedings of 13th International Workshop on Algebraic and Combinatorial Coding Theory (ACCT 2012), Pomorie, Bulgaria (2012) 257–262.
T. Maruta, Construction of optimal linear codes by geometric puncturing, Serdica J. Computing 7(1) (2013) 73–80.
T. Maruta, Griesmer bound for linear codes over finite fields, online resource, http://mars39.lomo.jp/opu/griesmer.htm.
T. Maruta and Y. Oya, On optimal ternary linear codes of dimension 6, Adv. Math. Commun. 5(3) (2011) 505–520.
T. Maruta, M. Shinohara, and M. Takenaka, Constructing linear codes from some orbits of projectivities, Discrete Math. 308(5-6) (2008) 832–841.
T. Maruta, T. Tanaka, and H. Kanda, Some generalizations of extension theorems for linear codes over finite fields, Australas. J. Combin. 60(2) (2014) 150–157.
T. Maruta and Y. Yoshida, A generalized extension theorem for linear codes, Des. Codes Cryptogr. 62(1) (2012) 121–130.
M. Takenaka, K. Okamoto, and T. Maruta, On optimal non-projective ternary linear codes, Discrete Math. 308(5-6) (2008) 842–854.
Y. Yoshida and T. Maruta, An extension theorem for $[n,k,d]_q$ codes with $\gcd(d,q)=2$, Australas. J. Combin. 48 (2010) 117–131.
