Nonexistence of some Griesmer codes over ${\mathbb{F}}_q$

Abstract

We prove the nonexistence of ${[g_q(4,d),4,d]}_q$ codes for $d=q^3/2-q^2-2q+1$ for $q=2^h$, $h\ge 3$, and for $d=2q^3-3q^2-2q+1$ for $q\ge 5$, where $g_q(k,d)={\sum }_{i=0}^{k-1}\lceil d/q^i\rceil$. We also prove the nonexistence of ${[g_q(k,d),k,d]}_q$ code for $d=(k-2)q^{k-1}-r{q}^{k-2}-q+1$ for $3\le k-1\le r\le q-q/p$, $q=p^h$ with $p$ prime.