New classes of NMDS codes with dimension 3

The singleton defect of an [n, k, d] linear code $\mathcal{C}$ is defined as $s(\mathcal{C})=n-k+1-d$. Codes with $s(\mathcal{C})=s({\mathcal{C}}^{\perp })=1$ are called near maximum distance separable (NMDS) codes. It is known that an $[n,3,n-3]$ NMDS code is equivalent to an (n, 3)-arc in PG(2, q). In this paper, by adding some suitable projective points into some known $(q+5,3)$-arcs in PG(2, q), we obtain two families of $[q+7,3,q+4]$ NMDS codes for even prime power q and a family of $[q+6,3,q+3]$ NMDS codes for odd prime power q. In addition, when $q=2^m$ and m is odd, by adding m suitable projective points into the maximum arcs in PG(2, q), we obtain a family of $[q+m+2,3,q+m-1]$ NMDS codes over ${\mathbb{F}}_q$, from which we further induce a family of NMDS codes with parameters $[q^t+m+2,3,q^t+m-1]$ over the extension field ${\mathbb{F}}_{q^t}$ for any odd integer t. All the resulting NMDS codes in this paper are shown to be linearly inequivalent to the NMDS codes constructed from elliptic curves, and their weight distributions are completely determined.