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Let q be a prime power and F<sub>q</sub> be the finite field of size q. In this paper we provide a Galois theoretical framework that allows to produce good polynomials for the Tamo and Barg construction of optimal locally recoverable codes (LRC). Using our approach we construct new good polynomials and therefore optimal LRCs with new parameters. The existing theory of good polynomials fits entirely in our new framework. The key advantage of our method is that we do not need to rely on arithmetic properties of the pair (q, r), where r is the locality of the code.

Constructions of Locally Recoverable Codes Which are Optimal