Locally recoverable codes in Hermitian function fields with certain types of divisors

A locally recoverable code with locality $\mathbf{r}$ can recover the missing coordinate from at most $\mathbf{r}$ symbols. The locally recoverable codes have attracted a lot of attention because they are more advanced coding techniques that are applied to distributed and cloud storage systems. In this work, we focus on locally recoverable codes in Hermitian function fields over ${\mathbb{F}}_{q^2}$, where $q$ is a prime power. With a certain type of divisor, we obtain an improved lower bound of the minimum distance for locally recoverable codes in Hermitian function fields. For doing this, we give explicit formulae of the dimension for some divisors of Hermitian function fields. We also present a standard that tells us when a divisor with certain places suggests an improved lower bound.