Functional repair codes: a view from projective geometry

Storage codes are used to ensure reliable storage of data in distributed systems; functional repair codes have the additional property that individual storage nodes that fail may be repaired efficiently, preserving the ability to recover original data and to further repair failed nodes. In this paper we show that the existing predominant coding theoretic and vector space models of repair codes can be given a unified treatment in a projective geometric framework, which permits a natural treatment of results such as the cutset bound. We find that many of the constructions proposed in the literature can be seen to arise from well-studied geometric objects, and that this perspective provides opportunities for generalisations and new constructions that can lead to greater flexibility in trade-offs between various desirable properties. We use this framework to explore the notion of strictly functional repair codes, for which there exist nodes that cannot be replaced exactly, and discuss how strict functionality can arise. We also consider the issue that the view of a repair code as a collection of sets of vector/projective subspaces is recursive in nature and makes it hard to discern when a collection of nodes forms a repair code. We provide another view using directed graphs that gives us non-recursive criteria for determining whether a family of collections of subspaces constitutes a functional, exact, or strictly functional repair code, which may be of use in searching for new codes with desirable properties.