LRC codes over characteristic $2$

In this work the construction of LRC codes given in [6] is completed, in the case of even characteristic. A general construction is presented, that enables us to obtain linear LRC codes of large length $n\approx q^4$, dimension and distance of order $q^4$, and locality $r=q-1$. In addition, the cases $q=4$ and $q=8$ are studied.