Minimum distance of elliptic codes (Q1191728)

The author examine three series of linear codes arising from elliptic curves. An \((n,k,d)\) code defined over an elliptic curve is almost maximum distance separable one that is \(d=n-k+1\) or \(d=n-k\). One of the most important properties of an elliptic function field defined over a fixed finite field \(Q\) is the fact that there is a bijection from its class group onto its set of prime divisors of degree one. This map is used in this paper to compute the minimal distance of the above-mentioned family of codes. In almost all cases these codes are not maximum distance separable, so \(d=n-k\). It is interesting that the MDS codes one obtains in the ``trivial'' cases belong to the class of generalized Reed-Solomon codes.