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 $\C$. This extended code, denoted by $\overline{\C}(-\bone)$ 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 is one of the two extending techniques for linear codes in the literature. 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 this type of extended linear codes. The second objective is to study this type of extended codes of a number of families of linear codes, including cyclic codes and nonbinary Hamming codes. Four families of distance-optimal or dimension-optimal linear codes are obtained with this extending technique. The parameters of certain extended codes of many families of linear codes are settled in this paper.

翻译：经典扩展$[n, k, d]$线性码$\C$的方法是在每个码字上添加一个全局奇偶校验坐标。该扩展码记作$\overline{\C}(-\bone)$，称为$\C$的标准扩展码，其参数为$[n+1, k, \bar{d}]$，其中$\bar{d}=d$或$\bar{d}=d+1$。这是文献中关于线性码的两种扩展技术之一。一些二元线性码族的标准扩展码已得到一定程度的研究。然而，对于非二元码的标准扩展码知之甚少。例如，非二元汉明码标准扩展码的最小距离问题已悬置七十余年。本文的首要目标是引入线性码的非标准扩展码，并建立此类扩展线性码的一般理论。第二个目标是研究包括循环码与非二元汉明码在内的若干线性码族的此类扩展码。通过该扩展技术，获得了四类距离最优或维数最优的线性码。本文确定了多个线性码族中某些扩展码的参数。