On the main conjecture on geometric MDS codes

For a linear [n, k, d] code, it is well known that d ≤ n−k+1 (see [12]), and when d = n−k+1, it is called the MDS code (maximum distance separable). For every n ≤ q + 1 there is an MDS [n, k, d] code for any given k and d satisfying d = n − k + 1. It is just the geometric codes on a rational curve (see [15]). There is a long-standing conjecture about the MDS linear codes that is called the main conjecture on MDS codes (see [12]): for a linear [n, k, d] MDS code, the length n ≤ q + 1 except q is even and k = 3 or k = q − 1 (in all cases 1 < k < n−1, i.e., exclude all trivial cases). This conjecture has been proved when q ≤ 11 or k ≤ 5 by the use of finite geometries. This conjecture for geometric codes has received considerable attention. In [9] Katsman and Tsfasman proved that this conjecture is true for elliptic MDS codes, and in [13] Munuera proved this conjecture is true in the case of geometric codes on genus-two curves, in which case the cardinality of Fq is required to be greater than 83. In [11] and [14] a bound for the length of MDS elliptic codes is studied. In this communication we announce the following two results and give a sketch of the proofs (see [2], [4], [3]).