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