Improving the dimension bound of Hermitian Lifted Codes

In this article we improve the dimension and minimum distance bound of the the Hermitian Lifted Codes LRCs construction from López, Malmskog, Matthews, Piñero and Wooters (López et. al.) via elementary univariarte polynomial division. They gave an asymptotic rate estimate of $0.007$. For the case where $q$ is a power of $2$ we improve the rate estimate to $0.010$ using univariate polynomial division.