Four new families of NMDS codes with dimension 4 and their applications

For an ${[n,k,d]}_q$ linear code $C$, the singleton defect of $C$ is defined by $S\left(C\right)=n-k+1-d$. When $S\left(C\right)=S\left(C^{\perp }\right)=1$, the code $C$ is called a near maximum distance separable (NMDS) code, where $C^{\perp }$ is the dual code of $C$. NMDS codes have important applications in finite projective geometries, designs and secret sharing schemes. In this paper, we present four new constructions of infinite families of NMDS codes with dimension 4 and completely determine their weight enumerators. As an application, we also determine the locality of the dual codes of these NMDS codes and obtain four families of distance-optimal and dimension-optimal locally recoverable codes.