ON THE MINIMUM LENGTH OF SOME LINEAR CODES OF DIMENSION 6

Title & Authors
ON THE MINIMUM LENGTH OF SOME LINEAR CODES OF DIMENSION 6
Cheon, Eun-Ju; Kato, Takao;

Abstract
For $\small{q^5-q^3-q^2-q+1{\leq}d{\leq}q^5-q^3-q^2}$, we prove the non-existence of a $\small{[g_q(6,d),6,d]_q}$ code and we give a $\small{[g_q(6,d)+1,6,d]_q}$ code by constructing appropriate 0-cycle in the projective space, where $\small{g_q (k,d)={{\sum}^{k-1}_{i=0}}{\lceil}\frac{d}{q^i}{\rceil}}$. Consequently, we have the minimum length $\small{n_q(6,d)=g_q(6,d)+1\;for\;q^5-q^3-q^2-q+1{\leq}d{\leq}q^5-q^3-q^2\;and\;q{\geq}3}$.
Keywords
Griesmer bound;linear code;0-cycle;minimum length;projective space;
Language
English
Cited by
1.
SOFT WS-ALGEBRAS,;;;

대한수학회논문집, 2008. vol.23. 3, pp.313-324
2.
DETERMINATION OF MINIMUM LENGTH OF SOME LINEAR CODES,;

충청수학회지, 2013. vol.26. 1, pp.147-159
1.
DETERMINATION OF MINIMUM LENGTH OF SOME LINEAR CODES, Journal of the Chungcheong Mathematical Society, 2013, 26, 1, 147
References
1.
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), no. 1-3, 229-268

2.
N. Hamada and T. Helleseth, The nonexistence of some ternary linear codes and update of the bounds for n3(6, d), 1 $\leq$ d $\leq$ 243, Math. Japon. 52 (2000), no. 1, 31-43

3.
R. Hill, Optimal linear codes, Cryptography and coding, II (Cirencester, 1989), 75-104, Inst. Math. Appl. Conf. Ser. New Ser., 33, Oxford Univ. Press, New York, 1992

4.
T. Maruta, On the nonexistence of q-ary linear codes of dimension five, Des. Codes Cryptogr. 22 (2001), no. 2, 165-177

5.
T. Maruta, Griesmer bound for linear codes over finite fields, Available: http://www. geocities.com/mars39.geo/griesmer.htm