Construction of Optimal Locally Repairable Codes via Automorphism Groups of Rational Function Fields

Locally repairable codes, or locally recoverable codes (LRC for short), are designed for applications in distributed and cloud storage systems. Similar to classical block codes, there is an important bound called the Singleton-type bound for locally repairable codes. In this paper, an optimal locally repairable code refers to a block code achieving this Singleton-type bound. Like classical MDS codes, optimal locally repairable codes carry some very nice combinatorial structures. Since the introduction of the Singleton-type bound for locally repairable codes, people have put tremendous effort into construction of optimal locally repairable codes. There are a few constructions of optimal locally repairable codes in the literature. Most of these constructions are realized via either combinatorial or algebraic structures. In this paper, we apply automorphism group of the rational function field to construct optimal locally repairable codes by considering the group action on projective lines over finite fields. Due to various subgroups of the projective general linear group, we are able to construct optimal locally repairable codes with flexible locality as well as smaller alphabet size comparable to the code length. In particular, we produce new families of q-ary locally repairable codes, including codes of length q+1 via cyclic groups.