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.

翻译：对于域 $\gf(q)$ 上长度为 $n$ 的线性码 $\C$ 及非零向量 $\bu \in \gf(q)^n$，Sun、Ding 和 Chen 定义了 $\C$ 的扩展线性码 $\overline{\C}(\bu)$，这是经典扩展码 $\overline{\C}(-\bone)$ 的推广，并称之为 $\C$ 的第二类扩展码（参见 arXiv:2307.04076 和 arXiv:2307.08053）。他们建立了扩展码 $\overline{\C}(\bu)$ 的一般理论，并研究了若干线性码族（包括循环码、射影二重权码、非二元汉明码以及一类可逆MDS循环码）的扩展码 $\overline{\C}(\bu)$。本文旨在研究有限域上 MDS 码 $\C$ 的扩展码 $\overline{\C}(\bu)$。主要结果表明：MDS $[n,k]$ 码 $\C$ 的扩展码 $\overline{\C}(\bu)$ 仍为 MDS 码当且仅当覆盖半径 $\rho(\mathcal{C}^{\bot})=k$ 且向量 $\bu$ 是对偶码 $\C^\perp$ 的深洞。基于这一主要结果，本文进一步研究了广义里德-所罗门（GRS）码及其扩展码的扩展码性质，并确定了若干 MDS 码族的覆盖半径。