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The main results of this paper are derived by using only simple Grobner basis techniques. We present a new construction of evaluation codes from Miura-Kamiya curves C"a"b. We estimate the minimum distance of the codes and estimate the minimum distance of a class of related one-point geometric Goppa codes. With respect to these estimates the new codes perform at least as well as the related geometric Goppa codes. In particular we consider codes from norm-trace curves. We show that our estimates give actually the true minimum distance of these codes. The new codes from norm-trace curves perform rather well. In many cases much better than the corresponding geometric Goppa codes. It turns out that an alternative description of the new codes from norm-trace curves can be made by using Hoholdt et al.'s in: V.S. Pless, W.C. Huffman (Eds.), Handbook of Coding Theory, Vol. 1, Elsevier, Amsterdam, 1998, pp. 871-961 (Chapter 10) construction of improved dual codes.