A New Approach to the Main Conjecture on Algebraic-Geometric MDS Codes

Abstract

The Main Conjecture on MDS Codes statesthat for every linear [n, k] MDS code over ${\mathbb{F}}$ q, if 1 <k < q, then n ≤ q+1,except when q is even and k=3 or k=q-1,in which cases n ≤ q +2. Recently, there has beenan attempt to prove the conjecture in the case of algebraic-geometriccodes. The method until now has been to reduce the conjectureto a statement about the arithmetic of the jacobian of the curve,and the conjecture has been successfully proven in this way forelliptic and hyperelliptic curves. We present a new approachto the problem, which depends on the geometry of the curve afteran appropriate embedding. Using algebraic-geometric methods,we then prove the conjecture through this approach in the caseof elliptic curves. In the process, we prove a new result aboutthe maximum number of points in an arc which lies on an ellipticcurve.