Non-existence of some 4-dimensional Griesmer codes over finite fields

Yıl 2018, , 101 - 116, 28.05.2018

Kazuki Kumegawa Tatsuya Maruta

https://doi.org/10.13069/jacodesmath.427968

Cited By: 3

Öz

We prove the non--existence of $[g_q(4,d),4,d]_q$ codes for $d=2q^3-rq^2-2q+1$ for $3 \le r \le (q+1)/2$, $q \ge 5$;
$d=2q^3-3q^2-3q+1$ for $q \ge 9$; $d=2q^3-4q^2-3q+1$ for $q \ge 9$; and $d=q^3-q^2-rq-2$ with $r=4, 5$ or $6$ for $q \ge 9$, where $g_q(4,d)=\sum_{i=0}^{3} \left\lceil
d/q^i \right\rceil$. This yields that $n_q(4,d) = g_q(4,d)+1$ for
$2q^3-3q^2-3q+1 \le d \le 2q^3-3q^2$,
$2q^3-5q^2-2q+1 \le d \le 2q^3-5q^2$ and
$q^3-q^2-rq-2 \le d \le q^3-q^2-rq$ with $4 \le r \le 6$ for $q \ge 9$
and that $n_q(4,d) \ge g_q(4,d)+1$ for
$2q^3-rq^2-2q+1 \le d \le 2q^3-rq^2-q$ for $3 \le r \le (q+1)/2$, $q \ge 5$ and
$2q^3-4q^2-3q+1 \le d \le 2q^3-4q^2-2q$ for $q \ge 9$,
where $n_q(4,d)$ denotes the minimum length $n$ for which
an $[n,4,d]_q$ code exists.

Anahtar Kelimeler

Optimal linear codes, Griesmer bound, Arcs in PG$(rq)$

Kaynakça

• [1] S. Ball, Table of bounds on three dimensional linear codes or (n,r)–arcs in PG(2, q), 2018, available at https://mat-web.upc.edu/people/simeon.michael.ball/codebounds.html
• [2] A. Bonisoli, Every equidistant linear code is a sequence of dual Hamming codes, Ars Combin. 18 (1984) 181–186.
• [3] E. J. Cheon, The non–existence of Griesmer codes with parameters close to codes of Belov type, Des. Codes Cryptogr. 61(2) (2011) 131–139.
• [4] E. J. Cheon, T. Maruta, On the minimum length of some linear codes, Des. Codes Cryptogr. 43(2–3) (2007) 123–135.
• [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, An extension theorem for linear codes, Des. Codes Cryptogr. 17(1–3) (1999) 151–157.
• [7] R. Hill, P. Lizak, Extensions of linear codes, Proc. 1995 IEEE International Symposium on Information Theory, 1995, p. 345.
• [8] J. W. P. Hirschfeld, Projective Geometries over Finite Fields, Second Edition, Clarendon Press, Oxford, 1998.
• [9] J. W. P. Hirschfeld, L. Storme, The packing problem in statistics, coding theory and finite projective spaces: update 2001, in: A. Blokhuis et al. (eds.), Finite Geometries, Developments in Mathematics Vol. 3, Springer, (2001) 201–246.
• [10] Y. Kageyama, T. Maruta, On the geometric constructions of optimal linear codes, Des. Codes Cryptogr. 81(3) (2016) 469–480.
• [11] R. Kanazawa, T. Maruta, On optimal linear codes over $F_8$, Electron. J. Combin. 18(1) (2011) #P34.
• [12] K. Kumegawa, T. Maruta, Non–existence of some Griesmer codes over $F_q$, Discrete Math. 339(2) (2016) 515–521.
• [13] 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.
• [14] I. N. Landjev, Optimal linear codes of dimension 4 over GF(5), Lecture Notes in Comp. Science 1255 (1997) 212–220.
• [15] I. N. Landjev, T. Maruta, On the minimum length of quaternary linear codes of dimension five, Discrete Math. 202(1–3) (1999) 145–161.
• [16] I. Landjev, A. Rousseva, The non–existence of (104, 22; 3, 5)–arcs, Adv. Math. Commun. 10(3) (2016) 601–611.
• [17] T. Maruta, On the achievement of the Griesmer bound, Des. Codes Cryptogr. 12(1) (1997) 83–87.
• [18] T. Maruta, On the minimum length of q–ary linear codes of dimension four, Discrete Math. 208–209 (1999) 427–435.
• [19] T. Maruta, On the non–existence of q–ary linear codes of dimension five, Des. Codes Cryptogr. 22(2) (2001) 165–177.
• [20] T. Maruta, A new extension theorem for linear codes, Finite Fields Appl. 10(4) (2004) 674–685.
• [21] T. Maruta, Construction of optimal linear codes by geometric puncturing, Serdica J. Comput. 7(1) (2013) 73–80.
• [22] T. Maruta, Griesmer bound for linear codes over finite fields, 2018, available at http://www.mi.s.osakafu-u.ac.jp/~maruta/griesmer/
• [23] T. Maruta, I. N. Landjev, A. Rousseva, On the minimum size of some minihypers and related linear codes, Des. Codes Cryptogr. 34(1) (2005) 5–15.
• [24] 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.
• [25] T. Maruta, Y. Oya, On optimal ternary linear codes of dimension 6, Adv. Math. Commun. 5(3) (2011) 505–520.
• [26] T. Maruta, Y. Yoshida, A generalized extension theorem for linear codes, Des. Codes Cryptogr. 62(1) (2012) 121–130.
• [27] M. Takenaka, K. Okamoto, T. Maruta, On optimal non–projective ternary linear codes, Discrete Math. 308(5–6) (2008) 842–854.
• [28] Y. Yoshida, T. Maruta, An extension theorem for [n, k, d]q codes with gcd(d, q) = 2, Aust. J. Combin. 48 (2010) 117–131.