On the minimum length of some linear codes of dimension $6$
Bull. Korean Math. Soc. 2008 Vol. 45, No. 3, 419-425
Published online September 1, 2008
Eun Ju Cheon and Takao Kato
Gyeongsang National University, Yamaguchi University
Abstract : For $q^5-q^3-q^2-q+1\le d \le q^5-q^3-q^2$, we prove the non-existence of a $[g_q(6,d),6,d]_q$ code and we give a $[g_q(6,d) +1, 6, d]_q$ code by constructing appropriate 0-cycle in the projective space, where $g_q(k,d)=\sum_{i=0}^{k-1}{\lceil{\frac{d}{q^i}}\rceil}$. Consequently, we have the minimum length $n_q(6,d)=g_q(6,d) +1$ for $q^5-q^3-q^2-q+1 \le d \le q^5 -q^3 -q^2$ and $q \ge 3$.
Keywords : Griesmer bound, linear code, $0$-cycle, minimum length, projective space
MSC numbers : 94B65, 94B05, 51E20, 05B25
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