On the minimum length of \(q\)-ary linear codes of dimension five (Q1360280)

A fundamental problem in coding theory is to determine \(n_q (k,d)\), the minimum value of \(n\) for which there exists an \([n,k,d]_q\) code over \(GF(q)\) for given \(q,k\) and \(d\). A lower bound on \(n_q(k,d)\) has been provided by the well known Griesmer bound. The purpose of this paper is to determine \(n_q(5,d)\) for \[ q^4-q^3-q -\sqrt q-2<d\leq q^4-q^4+ q^2-q \text{ for all }q. \] The technique employed in determining \(n_q(5,d)\) is a geometric method.