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The minimum distance is one of the most important combinatorial characterizations of a code. The maximum-likelihood decoding problem is one of the most important algorithmic problems of a code. While these problems are known to be hard for general linear codes, the techniques used to prove their hardness often rely on the construction of artificial codes. In general, much less is known about the hardness of the specific classes of natural linear codes. In this correspondence, we show that both problems are NP-hard for algebraic geometry codes. We achieve this by reducing a well-known NP-complete problem to these problems using a randomized algorithm. The family of codes in the reductions is based on elliptic curves. They have positive rates, but the alphabet sizes are exponential in the block lengths.