The extended codes of NMDS codes

For an [ n , k , d ] q linear code C , the singleton defect of C is defined by S ( C ) = n − k + 1 − d . If S ( C ) = S ( C ⊥ ) = 1 , C is called a near maximum distance separable (NMDS) code, where C ⊥ is the dual of C . NMDS codes have important applications in finite projective geometries, designs and secret sharing schemes. For a given linear code C of length n over F q and a nonzero vector u ∈ F q n , Sun, Ding and Chen [44] defined an extended linear code C ‾ ( u ) of C , which is a generalisation of the classical extended code C ‾ ( -1 ) of C . In this paper, for an [ n , k ] q NMDS code C and a nonzero vector u ∈ F q n , we provide a sufficient and necessary condition for the extended code C ‾ ( u ) to be an [ n + 1 , k ] q NMDS code. As an application, we construct two classes of NMDS codes of length q + 3 by extending the Roth-Lempel codes of length q + 2 . The weight distribution of these two classes of NMDS codes are also given. In addition, more NMDS codes are obtained from the NMDS code C ‾ ( u ) , and the covering radius and deep holes of some Roth-Lempel codes are determined. Moreover, we prove that the constructed NMDS codes are optimal locally recoverable codes.