Contribution to Munuera´s problem on the main conjecture of geometric hyperelliptic MDS codes

In coding theory, it is of great interest to know the maximal length of MDS codes. In fact, the main conjecture says that the length of MDS codes over F_q is less than or equal to q+1 (except for some special cases). Munuera (see ibid., vol.38, p.1573-7, 1992) proposed a new way to attack the main conjecture on MDS codes for geometric codes. In particular, he proved the conjecture for codes arising from curves of genus one or two when the cardinal of the ground field is large enough. He also asked whether a similar theorem can be proved for any hyperelliptic curve. The purpose of this correspondence is to give an affirmative answer. In fact, our method also proves the main conjecture for geometric MDS codes for q=2 if the genus of the hyperelliptic curve is either 1, 2 or 3, and for q=3 if the genus of the curve is 1