On the Nonexistence of $q$-ary Linear Codes Attaining the Griesmer Bound

Abstract

In this paper, we will prove that there exist no $[n,k,d{]}_q$ codes of $s{q}^{k-1}-(s+t)q^{k-2}-q^{k-4}\le d\le s{q}^{k-1}-(s+t)q^{k-2}$ attaining the Griesmer bound with $k\ge 4,1\le s\le k-2,t\ge 1$, and $s+t\le (q+1)\mathrm{\setminus }2$. Furthermore, we will prove that there exist no $[n,k,d{]}_q$ codes for $s{q}^{k-1}-(s+t)q^{k-2}-q^{k-3}\le d\le s$ attaining the Griesmer bound with $k\ge 3$, $1\le s\le k-2$, $t\ge 1$, and $s+t\le \sqrt{q}-1$. The results generalize the nonexistence theorems of Tatsuya Maruta (see $[7]$) and Andreas Klein (see $[4]$) to a larger class of codes.