On Maximally Recoverable Codes for Product Topologies
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Given a topology of local parity-check constraints, a maximally recoverable code (MRC) can correct all erasure patterns that are information-theoretically correctable. In a grid-like topology, there are a local constraints in every column forming a column code, b local constraints in every row forming a row code, and h global constraints in an (m × n) grid of codeword. Recently, Gopalan et al. initiated the study of MRCs under grid-like topology, and derived a necessary and sufficient condition, termed as the regularity condition, for an erasure pattern to be recoverable when a=1, h=0. In this paper, we consider MRCs for product topology (h=0). First, we construct a certain bipartite graph based on the erasure pattern satisfying the regularity condition for product topology (any a, b, h=0) and show that there exists a complete matching in this graph. We then present an alternate direct proof of the sufficient condition when a=1, h=0. We later extend our technique to study the topology for a=2, h=0, and characterize a subset of recoverable erasure patterns in that case. For both a=1, 2, our method of proof is uniform, i.e., by constructing tensor product G_col⊗ G_row of generator matrices of column and row codes such that certain square sub-matrices retain full rank. The full-rank condition is proved by resorting to the matching identified earlier and also another set of matchings in erasure sub-patterns.
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