Near-MDS codes arising from algebraic curves

Every elliptic quartic Γ4 of PG(3,q) with nGF(q)-rational points provides a near-MDS code C of length n and dimension 4 such that the collineation group of Γ4 is isomorphic to the automorphism group of C. In this paper we assume that GF(q) has characteristic p>3. We classify the linear collineation groups of PG(3,q) which can preserve an elliptic quartic of PG(3,q). Also, we prove for q⩾113 that if the j-invariant of Γ4 does not disappear, then C cannot be extended in a natural way by adding a point of PG(3,q) to Γ4.