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.

We suggest a Jacobi form over a number field $\mathbb{Q}(\sqrt{5},i)$; for obtaining this, we use a linear code $C$ over $R:={\mathbb{F}}_4+u{\mathbb{F}}_4$, where $u^2=0$. We introduce MacWilliams identities for both complete weight enumerator and symmetrized weight enumerator in higher genus $g\ge 1$ of a linear code over $R$. Finally, we give invariants via a self-dual code of even length over $R$.