Classification of the $[n,3,n-3{]}_q$ NMDS Codes Over $GF(7)$, $GF(8)$ and $GF(9)$

Abstract

A linear $[n,k,d{]}_q$ code $C$ is called NMDS if $d(C)=n–k$ and $d(C^{\perp })=k$. In this paper, the classification of the $[n,3,n-k{]}_q$ NMDS codes is given for $q=7,8,9$. It has been found using the correspondence between $[n,3,n-k{]}_q$ NMDS codes and $(n,3)$-arcs of $\mathrm{P}\mathrm{G}(2,q)$.