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Repair Locality From a Combinatorial Perspective

Repair locality is a desirable property for erasure codes in distributed storage systems. Recently, different structures of local repair groups have been proposed in the definitions of repair locality. In this paper, the concept of regenerating set is introduced to characterize the local repair groups. A definition of locality $r^{(δ-1)}$ (i.e., locality $r$ with repair tolerance $δ-1$) under the most general structure of regenerating sets is given. All previously studied locality turns out to be special cases of this definition. Furthermore, three representative concepts of locality proposed before are reinvestigated under the framework of regenerating sets, and their respective upper bounds on the minimum distance are reproved in a uniform and brief form. Additionally, a more precise distance bound is derived for the square code which is a class of linear codes with locality $r^{(2)}$ and high information rate, and an explicit code construction attaining the optimal distance bound is obtained.