Extended codes and deep holes of MDS codes

For a given linear code $\C$ of length $n$ over $\gf(q)$ and a nonzero vector $\bu$ in $\gf(q)^n$, Sun, Ding and Chen defined an extended linear code $\overline{\C}(\bu)$ of $\C$, which is a generalisation of the classical extended code $\overline{\C}(-\bone)$ of $\C$ and called the second kind of an extended code of $\C$ (see arXiv:2307.04076 and arXiv:2307.08053). They developed some general theory of the extended codes $\overline{\C}(\bu)$ and studied the extended codes $\overline{\C}(\bu)$ 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 $\overline{\C}(\bu)$ of MDS codes $\C$ over finite fields. The main result of this paper is that the extended code $\overline{\C}(\bu)$ of an MDS $[n,k]$ code $\C$ remains MDS if and only if the covering radius $\rho(\mathcal{C}^{\bot})=k$ and the vector $\bu$ is a deep hole of the dual code $\C^\perp$. As applications of this main result, the extended codes of the GRS codes and extended GRS codes are investigated and the covering radii of several families of MDS codes are determined.