When Does the Extended Code of an MDS Code Remain MDS?

For a given linear code $\mathcal {C}$ of length n over ${\mathrm {GF}}(q)$ and a nonzero vector u in ${\mathrm {GF}}(q)^{n}$ , Sun, Ding and Chen defined an extended linear code $\overline {\mathcal {C}}({\mathbf {u}})$ of $\mathcal {C}$ , which is a generalisation of the classical extended code $\overline {\mathcal {C}}(-{\mathbf {1}})$ of $\mathcal {C}$ and called the second kind of an extended code of $\mathcal {C}$ (see Finite Fields Appl., vol. 96, 102401, 2024 and Discrete Math., vol. 347, no. 9, 114080, 2024). They developed some general theory of the extended codes $\overline {\mathcal {C}}({\mathbf {u}})$ and studied the extended codes $\overline {\mathcal {C}}({\mathbf {u}})$ of several families of linear codes, including cyclic codes, projective two-weight codes, nonbinary Hamming codes, and a family of reversible MDS cyclic codes. The objective of this paper is to investigate the extended codes $\overline {\mathcal {C}}({\mathbf {u}})$ of MDS codes $\mathcal {C}$ over finite fields. The main result of this paper is that the extended code $\overline {\mathcal {C}}({\mathbf {u}})$ of an MDS $[n,k]$ code $\mathcal {C}$ remains MDS if and only if the covering radius $\rho (\mathcal {C}^{\bot })=k$ and the vector u is a deep hole of the dual code ${\mathcal {C}}^{\perp } $ . As applications of this main result, an equivalent statement of MDS Conjecture is presented, the extended codes of the GRS codes and extended GRS codes are investigated, and the covering radii and some deep holes of several families of MDS codes are also determined.