Roth–Lempel NMDS Codes of Non-Elliptic-Curve Type

The defect of an$[n,k,d]$linear code is defined as$s({\mathcal{ C}})=n-k+1-d$. Codes with$s({\mathcal{ C}})=0$are called maximum distance separable (MDS), while codes with$s({\mathcal{ C}})=s({\mathcal{ C}}^{\perp})=1$are called near maximum distance separable (NMDS). NMDS codes correspond to interesting objects in finite geometry and have nice applications in combinatorics and cryptography. There have been many constructions of NMDS codes, but most of them are focus on fixed$q$or$k$, except for constructions from elliptic curves. Roth and Lempel (IEEE Trans. Inf. Theory 1989) constructed a type of linear codes (referred as Roth-Lempel codes), and presented the necessary and sufficient conditions of Roth-Lempel code to be MDS. Especially, they pointed out that the resultant MDS codes is not linearly equivalent to Reed-Solomn codes. In this paper, the NMDS properties of Roth-Lempel codes will be analyzed. We also obtain the necessary and sufficient condition of Roth-Lempel codes to be NMDS, and further completely determine the weight distributions of Roth-Lempel codes with length$q+2$and dimension$3\leq k\leq q$. Besides, by analyzing the upper bound for the code lengths of elliptic curve MDS codes, we illustrate the linearly inequivalence of Roth-Lempel NMDS codes and elliptic curve NMDS codes when their corresponding code lengths exceed$4(q+2\sqrt {q}+1)/5+1$.