In the last decade, Locally Recoverable Codes (LRC) have been a critical topic in communication and distributed storage. In addition, the minimum distance, dimension and length of a code LRCs also consider the locality of a code, i.e. the minimum number of entries needed to recover a given entry for any codeword. The parameters of LRCs are subject to a general Singleton bound and codes achieving the bound are called optimal LRCs. Constructions are known when the underlying field size of the code is larger than the length of the code. However, still little is known about the existence of optimal LRCs over small underlying field sizes. In this talk, I will show how we established new bounds that depend on locality and the field size of code using a duality theory of LRCs and the combinatorial structure of the code. This talk is based on joint work with A. Gruica and A. Ravagna
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