In this work, we study the computational complexity of the Minimum Distance Code Detection problem. In this problem, we are given a set of noisy codeword observations and we wish to find a code in a set of linear codes C of a given dimension k, for which the sum of distances between the observations and the code is minimized. We prove that, for the practically relevant case when the set C only contains a fixed number of candidate linear codes, the detection problem is NP-hard and we identify a number of interesting open questions related to the code detection problem.