On the Computational Complexity of Blind Detection of Binary Linear Codes

TL;DR

This paper investigates the computational difficulty of identifying the correct binary linear code from noisy observations, proving NP-hardness for cases with a fixed set of candidate codes and highlighting open questions.

Contribution

It establishes the NP-hardness of the Minimum Distance Code Detection problem for a fixed set of candidate linear codes, advancing understanding of its computational complexity.

Findings

Detection problem is NP-hard for fixed candidate code sets

Identifies open questions in code detection complexity

Provides theoretical insights into code detection challenges

Abstract

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.