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

Constructions of near MDS codes which are optimal locally recoverable codes