A Comment on the Weight Structure of Generator Matrices of Linear Codes

We prove that for any linear $q$-ary $[n,k,d]$-code with $n=t+g_q(k,d)$, where $g_q(k,d)=\sum\limits_{j=0}^{k-1}\biggl\lceil\dfrac d{q^j}\biggr\rceil$ is the Griesmer function, a generator matrix can be selected from code vectors of weight not exceeding $d+t$.