MRHS Equation Systems that can be Solved in Polynomial Time

In this article we study the difficulty of solving Multiple Right-Hand Side (MRHS) equation systems. In the first part we show that, in general, solving MRHS systems is NP-hard. In the next part we focus on special (large) families of MRHS systems that can be solved in polynomial time with two algorithms: one based on linearisation of MRHS equations, and the second one based on decoding problems that can be solved in polynomial time.