  Because of the recent applications to distributed storage systems,
researchers have introduced a new class of block codes, i.e., locally
recoverable (LRC) codes. LRC codes can recover information from erasure(s) by
accessing a small number of erasure-free code symbols and increasing the
efficiency of repair processes in large-scale distributed storage systems. In
this context, Tamo and Barg first gave a breakthrough by cleverly introducing a
good polynomial notion. Constructing good polynomials for locally recoverable
codes achieving Singleton-type bound (called optimal codes) is challenging and
has attracted significant attention in recent years. This article aims to
increase our knowledge on good polynomials for optimal LRC codes. Using tools
from algebraic function fields and Galois theory, we continue investigating
those polynomials and studying them by developing the Galois theoretical
approach initiated by Micheli in 2019. Specifically, we push further the study
of a crucial parameter $\mathcal G(f)$ (of a given polynomial $f$), which
measures how much a polynomial is "good" in the sense of LRC codes. We provide
some characterizations of polynomials with minimal Galois groups and prove some
properties of finite fields where polynomials exist with a specific size of
Galois groups. We also present some explicit shapes of polynomials with small
Galois groups. For some particular polynomials $f$, we give the exact formula
of $\mathcal G(f)$.
