Optimal and Asymptotically Good Locally Repairable Codes via Propagation Rules

In classical coding theory, it is common to construct new codes via propagation rules. There are various propagation rules to construct classical block codes. However, propagation rules have not been extensively explored for locally repairable codes. In this paper, we systematically study some of propagation rules to construct optimal and asymptotically good locally repairable codes. To our surprise, these simple propagation rules produce interesting results. Firstly, by a lengthening propagation rule that adds some rows and columns to a parity-check matrix of a given linear code, we are able to convert a classical maximum distance separable (MDS) code into a Singleton-optimal locally repairable code and provide a simplified proof of the asymptotic Tafasman-Vladut-Zink bound which exceeds the asymptotic Gilbert-Varshamov bound of locally repairable codes. Secondly, by concatenating a locally repairable code as an inner code with a classical block code as an outer code, we obtain a family of dimension-optimal locally repairable codes. Thirdly, we can make use of the shortening technique to produce more dimension-optimal locally repairable codes. Finally, one of phenomena that we observe in this paper is that some trivial propagation rules in classical block codes do not hold anymore for locally repairable codes.