Coding Theory using Linear Complexity of Finite Sequences

We define a metric on $\mathbb{F}_q^n$ using the linear complexity of finite sequences. We will then develop a coding theory for this metric. We will give a Singleton-like bound and we will give constructions of subspaces of $\mathbb{F}_q^n$ achieving this bound. We will compute the size of balls with respect to this metric. In other words we will count how many finite sequences have linear complexity bounded by some integer $r$. The paper is motivated in part by the desire to design new code based cryptographic systems.