On the achievement of the Griesmer bound (Q1364213)

For \(d=(k-2)q^{k-1} -(k-1)q^{k-2}\), there does not exist a linear \((n,k)\)-code over \(GF(q)\) of minimum Hamming distance \(d\) which attains the Grieser bound for \(q\geq k\), \(k=3,4,5\) and for \(q\geq 2k-3\), \(k\geq 6\). For \(q\geq k\) and \(k=3,4\) this has been known before [\textit{S. M. Dodunekov}, C. R. Acad. Bulg. Sci. 39, 39-41 (1986; Zbl 0628.94014), and T. Maruta (1995)]. The remaining parts of the theorem are proved by considering the columns of a generator matrix of an \((n,k,d)_q\)-code as a multiset of \(n\) points in the projective space \(PG(k-1,q)\).