Near-MDS Codes from Maximal Arcs in PG$(2,q)$

TL;DR

This paper constructs new near-MDS codes from maximal arcs in projective planes, analyzes their properties, and shows they are both distance- and dimension-optimal locally recoverable codes.

Contribution

It introduces two classes of NMDS codes derived from maximal arcs in PG(2,q), detailing their weight distribution and locality, and establishing their optimality.

Findings

Constructed two classes of (q+5,3) NMDS codes from maximal arcs.

Determined the exact weight distribution of these NMDS codes.

Proved the codes and their duals are both distance- and dimension-optimal locally recoverable codes.

Abstract

The singleton defect of an $[n,k,d]$ linear code ${\cal C}$ is defined as $s({\cal C})=n-k+1-d$. Codes with $S({\cal C})=0$ are called maximum distance separable (MDS) codes, and codes with $S(\cal C)=S(\cal C ^{\bot})=1$ are called near maximum distance separable (NMDS) codes. Both MDS codes and NMDS codes have good representations in finite projective geometry. MDS codes over $F_q$ with length $n$ and $n$-arcs in PG$(k-1,q)$ are equivalent objects. When $k=3$, NMDS codes of length $n$ are equivalent to $(n,3)$-arcs in PG$(2,q)$. In this paper, we deal with the NMDS codes with dimension 3. By adding some suitable projective points in maximal arcs of PG$(2,q)$, we can obtain two classes of $(q+5,3)$-arcs (or equivalently $[q+5,3,q+2]$ NMDS codes) for any prime power $q$. We also determine the exact weight distribution and the locality of such NMDS codes and their duals. It turns out that the resultant NMDS codes and their duals are both distance-optimal and dimension-optimal locally recoverable codes.