  In this article we prove a Griesmer type bound for additive codes over finite
fields. This new bound gives an upper bound on the length of maximum distance
separable (MDS) codes, codes which attain the Singleton bound. We will also
consider codes to be MDS if they obtain the fractional Singleton bound, due to
Huffman. We prove that this bound in the fractional case can be obtained by
codes whose length surpasses the length of the longest known codes in the
non-fractional case. We also provide some exhaustive computational results over
small fields and dimensions.
