On optimal codes over the field with five elements (Q1404325)

Let \(n_q(k,d)\) denote the minimal length of a linear code of dimension \(k\) and minimum distance \(d\) over the field \(\text{GF}(q)\). An important problem in coding theory is to find the exact value of \(n_q(k,d)\), for given \(k\), \(d\) and \(q\). It is known that a full length linear code is equivalent to an arc in a finite projective space. The authors use this equivalence to prove the nonexistence of Griesmer codes over \(\text{GF}(5)\) for \(k=4\) and \(d=33,83,163,164\).