On the minimum length of linear codes of dimension 5

A fundamental problem in coding theory is to find the exact value $n_q(k,d)$, the minimum length n for which an ${[n,k,d]}_q$ code exists for given $q,k$ and d. The code of length $n_q(k,d)$ is called length optimal. Finding length optimal codes presents the most interesting problem in optimal linear codes, because length optimal codes are simultaneously distance optimal and dimension optimal. In this article, we focus on finding 5-dimensional length optimal codes. We prove the nonexistence of 5-dimensional Griesmer code, and it is proved $n_q(5,d)=g_q(5,d)+1$ for $3q^4-4q^3-aq+1\le d\le 3q^4-4q^3-q$ with $1\le a\le \lfloor \frac{2}{3}q+1\rfloor$ and $2q^4-2q^3-2q^2-q+1\le d\le 2q^4-2q^3-2q^2$ with $q\ge 5$.