Two classes of (r, t)-locally repairable codes

An (r, t)-locally repairable code satisfies a property that the value at each coordinate can be recovered from t disjoint repair sets each containing at most r other coordinates. This property is extremely useful in distributed storage systems for hot data. In this paper, we propose two constructions of (r, t)-LRCs. The first one is a cyclic code of which the parity check polynomial is closely related to the trace function over finite fields. This code can achieve high availability and large minimum distance. The second one is based on the inclusion matrix of linear subspaces in F
q
^m
. For some specific parameters, we prove that its information rate is always higher than r over r+t which was conjectured to be near to the optimal rate for (r, t)-LRCs (A. Wang and Z. Zhang, ISIT 2015). By shortening this code in a specially designed way, we obtain (r, t)-LRCs with more desirable locality r at a slight expense of information rate.