On the Nonexistence of \(q\)-ary Linear Codes Attaining the Griesmer Bound

Xiuli Li1,2
1Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China
2School of Math. and Phys., Qingdao University of Science and Technology, Qingdao 266061, China

Abstract

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