  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 $\mathcal{C}$
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 $\mathcal{C}$ 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.
