  We consider linear cyclic codes with the locality property, or locally
recoverable codes (LRC codes). A family of LRC codes that generalizes the
classical construction of Reed-Solomon codes was constructed in a recent paper
by I. Tamo and A. Barg (IEEE Transactions on Information Theory, no. 8, 2014;
arXiv:1311.3284). In this paper we focus on the optimal cyclic codes that arise
from the general construction. We give a characterization of these codes in
terms of their zeros, and observe that there are many equivalent ways of
constructing optimal cyclic LRC codes over a given field. We also study
subfield subcodes of cyclic LRC codes (BCH-like LRC codes) and establish
several results about their locality and minimum distance.
