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We prove a new lower bound on the field size of locally repairable codes(LRCs). Additionally, we construct maximally recoverable (MR) codes which arecyclic. While a known construction for MR codes has the same parameters, itproduces non-cyclic codes. Furthermore, we prove both necessary conditions andsufficient conditions that specify when the known non-cyclic MR codes may bepermuted to become cyclic, thus proving our construction produces cyclic MRcodes with new parameters. Furthermore, using our new bound on the field size,we show that the new cyclic MR codes have optimal field size in certain cases.Other known LRCs are also shown to have optimal field size in certain cases.

A Bound on the Minimal Field Size of LRCs, and Cyclic MR Codes That Attain It