Self-Dual Near MDS Codes from Elliptic Curves

In recent years, self-dual MDS codes have attracted a lot of attention due to theoretical interest and practical importance. Similar to self-dual MDS codes, self-dual near MDS (NMDS for short) codes have nice structures as well. From both theoretical and practical points of view, it is natural to study self-dual NMDS codes. Although there has been lots of work on NMDS codes in literature, little is known for self-dual NMDS codes. It seems more challenging to construct self-dual NMDS codes than self-dual MDS codes. The only work on construction of self-dual NMDS codes shows existence of q-ary self-dual NMDS codes of length q - 1 for odd prime power q or length up to 16 for some small primes q with q ≤ 197. In this paper, we make use of properties of elliptic curves to construct selfdual NMDS codes. It turns out that, as long as 2|q and n is even with 4 ≤ n ≤ q + 12√qJ - 2, one can construct a self-dual NMDS code of length n over Fq. Furthermore, for odd prime power q, there exists a self-dual NMDS code of length n over Fq if q ≥ 4
ⁿ⁺³
x (n + 3)
²
.