Non–existence of some 4–dimensional Griesmer codes over finite fields
DOI:
https://doi.org/10.13069/jacodesmath.427968Keywords:
Optimal linear codes, Griesmer bound, Arcs in PG(r, q)Abstract
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.
Received: 18 May 2016 Accepted: 25 April 2018Downloads
References
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.
A. Bonisoli, Every equidistant linear code is a sequence of dual Hamming codes, Ars Combin. 18 (1984) 181–186.
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.
E. J. Cheon and T. Maruta, On the minimum length of some linear codes, Des. Codes Cryptogr. 43(2–3) (2007) 123–135.
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, An extension theorem for linear codes, Des. Codes Cryptogr. 17(1–3) (1999) 151–157.
R. Hill and P. Lizak, Extensions of linear codes, In: Proc. 1995 IEEE International Symposium on Information Theory (ISIT), Whistler, BC, Canada (1995) 345.
J. W. P. Hirschfeld, Projective Geometries over Finite Fields, 2nd ed., Oxford Mathematical Monographs, Clarendon Press, Oxford, 1998.
J. W. P. Hirschfeld and 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, Boston, MA (2001) 201–246.
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) #P34.
K. Kumegawa and T. Maruta, Non–existence of some Griesmer codes over $\mathbb{F}_q$, Discrete Math. 339(2) (2016) 515–521.
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) #P1.50.
I. N. Landjev, Optimal linear codes of dimension 4 over $GF(5)$, In: T. Mora and H. Stichtenoth (Eds.), Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (AAECC-12), Lecture Notes in Comput. Sci., vol. 1255, Springer, Berlin, Heidelberg (1997) 212–220.
I. N. Landjev and T. Maruta, On the minimum length of quaternary linear codes of dimension five, Discrete Math. 202(1–3) (1999) 145–161.
I. Landjev and A. Rousseva, The non–existence of $(104, 22; 3, 5)$–arcs, Adv. Math. Commun. 10(3) (2016) 601–611.
T. Maruta, On the achievement of the Griesmer bound, Des. Codes Cryptogr. 12(1) (1997) 83–87.
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 non–existence 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, 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 (2018), available at http://www.mi.s.osakafu-u.ac.jp/~maruta/griesmer/.
T. Maruta, I. N. Landjev, and A. Rousseva, On the minimum size of some minihypers and related linear codes, Des. Codes Cryptogr. 34(1) (2005) 5–15.
T. Maruta, A. Kikui, and Y. Yoshida, On the uniqueness of $(48, 6)$–arcs in $PG(2, 9)$, Adv. Math. Commun. 3(1) (2009) 29–34.
T. Maruta and Y. Oya, On optimal ternary linear codes of dimension 6, Adv. Math. Commun. 5(3) (2011) 505–520.
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.
