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.

翻译：本文证明了有限域上加法码的Griesmer型界。这一新界为达到Singleton界的最大距离可分（MDS）码的长度提供了上界。根据Huffman的理论，若码达到分数Singleton界，我们亦将其视为MDS码。我们证明了在分数情形下，该界可通过长度超过非分数情形中已知最长码长度的码达到。此外，我们还提供了小域和小维数下的若干穷举计算结果。