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We consider locally repairable codes over small fields and propose constructions of optimal cyclic and linear codes in terms of the dimension for a given distance and length. Four new constructions of optimal linear codes over small fields with locality properties are developed. The first two approaches give binary cyclic codes with locality two. While the first construction has availability one, the second binary code is characterized by multiple available repair sets based on a binary Simplexdoi:10.1109/itw.2015.7133123 dblp:conf/itw/ZehY15 fatcat:r2un6f46n5bgngyvyi6prmjfam