On the Computational Complexity of Blind Detection of Binary Linear Codes

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.