Bulletin of the
Korean Mathematical Society BKMS

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