  We introduce the notion of Universally Decodable Matrices of Genus g (UDMG),
which for g=0 reduces to the notion of Universally Decodable Matrices (UDM)
introduced in [8]. A UDMG is a set of L matrices over a finite field, each with
K rows, and a linear independence condition satisfied by collections of K+g
columns formed from the initial segments of the matrices. We consider the
mathematical structure of UDMGs and their relation to linear vector codes. We
then give a construction of UDMG based on curves of genus g over the finite
field, which is a natural generalization of the UDM constructed in [8]. We
provide upper (and constructable lower) bounds for L in terms of K, q, g, and
the number of columns of the matrices. We will show there is a fundamental
trade off (Theorem 5.4) between L and g, akin to the Singleton bound for the
minimal Hamming distance of linear vector codes.
