A linear code with parameters $[n,k,n-k]$ is said to be almost maximum distance separable (AMDS for short). An AMDS code whose dual is also AMDS is referred to as an near maximum distance separable (NMDS for short) code. NMDS codes have nice applications in finite geometry, combinatorics, cryptography and data storage. In this paper, we first present several constructions of NMDS codes and determine their weight enumerators. In particular, some constructions produce NMDS codes with the same parameters but different weight enumerators. Then we determine the locality of the NMDS codes and obtain many families of distance-optimal and dimension-optimal locally repairable codes.

翻译：带有 $[n,k,n-k] 参数的线性代码据说几乎是最大距离可分离的(简称AMDS),一个双倍于AMDS的AMDS代码被称为近最大距离可分离的代码(简称NMDS), NMDS代码在有限的几何、组合式、加密和数据储存方面应用良好,在本文中,我们首先介绍若干NDS代码的构造,并确定其重量计数器,特别是一些建筑生产NMDS代码,其参数相同,但重量计数器不同。然后我们确定NMDS代码的位置,并获得许多距离最优和尺寸最优的本地可修理代码。