Near MDS codes from oval polynomials

Abstract A linear code with parameters of the form [ n , k , n − k + 1 ] is referred to as an MDS (maximum distance separable) code. A linear code with parameters of the form [ n , k , n − k ] is said to be almost MDS (i.e., almost maximum distance separable) or AMDS for short. A code is said to be near maximum distance separable (in short, near MDS or NMDS) if both the code and its dual are almost maximum distance separable. Near MDS codes correspond to interesting objects in finite geometry and have nice applications in combinatorics and cryptography. There are many unsolved problems about near MDS codes. It is hard to construct an infinite family of near MDS codes whose weight distributions can be settled. In this paper, seven infinite families of [ 2 m + 1 , 3 , 2 m − 2 ] near MDS codes over GF ( 2 m ) and seven infinite families of [ 2 m + 2 , 3 , 2 m − 1 ] near MDS codes over GF ( 2 m ) are constructed with special oval polynomials for odd m . In addition, nine infinite families of optimal [ 2 m + 3 , 3 , 2 m ] near MDS codes over GF ( 2 m ) are constructed with oval polynomials in general. The weight distributions of these near MDS codes are settled.