Construction of a 3-Dimensional MDS code

TL;DR

This paper presents a method for constructing three-dimensional MDS codes over finite fields using Hermitian forms in projective geometry, expanding the known parameters for such codes.

Contribution

It introduces a new construction technique for q-ary [N,3,N-2]-MDS codes using non-degenerate Hermitian forms in projective geometry.

Findings

Constructed MDS codes with parameters N ≤ q+1 for odd q

Constructed MDS codes with parameters N ≤ q+2 for even q

Provided explicit geometric construction methods

Abstract

In this paper, we describe a procedure for constructing $q$--ary $[N,3,N-2]$--MDS codes, of length $N\leq q+1$ (for $q$ odd) or $N\leq q+2$ (for $q$ even), using a set of non--degenerate Hermitian forms in $PG(2,q^2)$.