Abstract
The Main Conjecture on MDS Codes states that for every linear [n, κ] MDS code over \(\mathbb{F}\) q , if 1 < κ < q, then n ≤ q + 1, except when q is seven and κ = 3 or κ = q − 1, in which cases n ≤ 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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Communicated by: V. Tonchev
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Walker, J.L. A new approach to the main conjecture on algebraic-geometric MDS Codes. Des Codes Crypt 9, 115–120 (1996). https://doi.org/10.1007/BF00169780
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DOI: https://doi.org/10.1007/BF00169780