The classical way of extending an $[n, k, d]$ linear code $\C$ is to add an overall parity-check coordinate to each codeword of the linear code $\C$. The extended code, denoted by $\overline{\C}$ and called the standardly extended code of $\C$, is a linear code with parameters $[n+1, k, \bar{d}]$, where $\bar{d}=d$ or $\bar{d}=d+1$. This extending technique is one of the classical ways to construct a new linear code with a known linear code and a way to study the original code $\C$ via its extended code $\overline{\C}$. The standardly extended codes of some families of binary linear codes have been studied to some extent. However, not much is known about the standardly extended codes of nonbinary codes. For example, the minimum distances of the standardly extended codes of the nonbinary Hamming codes remain open for over 70 years. The first objective of this paper is to introduce the nonstandardly extended codes of a linear code and develop some general theory for extended linear codes. The second objective is to study the extended codes of several families of linear codes, including cyclic codes, projective two-weight codes and nonbinary Hamming codes. Many families of distance-optimal linear codes are obtained with the extending technique.

翻译：经典的对 $[n, k, d]$ 线性码 $\C$ 进行扩展的方法是在该线性码 $\C$ 的每个码字上添加一个全局奇偶校验坐标。扩展后的码记作 $\overline{\C}$，称为 $\C$ 的标准扩展码，它是一个参数为 $[n+1, k, \bar{d}]$ 的线性码，其中 $\bar{d}=d$ 或 $\bar{d}=d+1$。这种扩展技术是利用已知线性码构造新线性码的经典方法之一，也是通过扩展码 $\overline{\C}$ 研究原始码 $\C$ 的一种途径。对于某些二元线性码族的标准扩展码已有一定程度的研究。然而，关于非二元码的标准扩展码所知甚少。例如，非二元汉明码的标准扩展码的最小距离问题至今已悬而未决超过70年。本文的首要目标是介绍线性码的非标准扩展码，并发展扩展线性码的一般理论。第二个目标是研究若干线性码族的扩展码，包括循环码、射影二重权码和非二元汉明码。利用该扩展技术，本文获得了许多距离最优线性码族。