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 \(\mathrm{PG}(2,q)\).