7 research outputs found
Maximally Recoverable Codes with Hierarchical Locality
Maximally recoverable codes are a class of codes which recover from all
potentially recoverable erasure patterns given the locality constraints of the
code. In earlier works, these codes have been studied in the context of codes
with locality. The notion of locality has been extended to hierarchical
locality, which allows for locality to gradually increase in levels with the
increase in the number of erasures. We consider the locality constraints
imposed by codes with two-level hierarchical locality and define maximally
recoverable codes with data-local and local hierarchical locality. We derive
certain properties related to their punctured codes and minimum distance. We
give a procedure to construct hierarchical data-local MRCs from hierarchical
local MRCs. We provide a construction of hierarchical local MRCs for all
parameters. For the case of one global parity, we provide a different
construction of hierarchical local MRC over a lower field size.Comment: 6 pages, accepted to National Conference of Communications (NCC) 201
On Maximally Recoverable Codes for Product Topologies
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 local constraints in every
column forming a column code, local constraints in every row forming a row
code, and global constraints in an 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 .
In this paper, we consider MRCs for product topology (). First, we
construct a certain bipartite graph based on the erasure pattern satisfying the
regularity condition for product topology (any , ) and show that
there exists a complete matching in this graph. We then present an alternate
direct proof of the sufficient condition when . We later extend our
technique to study the topology for , and characterize a subset of
recoverable erasure patterns in that case. For both , our method of
proof is uniform, i.e., by constructing tensor product 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.Comment: 6 pages, accepted to National Conference of Communications (NCC) 201