103 research outputs found
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
Constructions of Maximally Recoverable Local Reconstruction Codes via Function Fields
Local Reconstruction Codes (LRCs) allow for recovery from a small number of erasures in a local manner based on just a few other codeword symbols. They have emerged as the codes of choice for large scale distributed storage systems due to the very efficient repair of failed storage nodes in the typical scenario of a single or few nodes failing, while also offering fault tolerance against worst-case scenarios with more erasures. A maximally recoverable (MR) LRC offers the best possible blend of such local and global fault tolerance, guaranteeing recovery from all erasure patterns which are information-theoretically correctable given the presence of local recovery groups. In an (n,r,h,a)-LRC, the n codeword symbols are partitioned into r disjoint groups each of which include a local parity checks capable of locally correcting a erasures. The codeword symbols further obey h heavy (global) parity checks. Such a code is maximally recoverable if it can correct all patterns of a erasures per local group plus up to h additional erasures anywhere in the codeword. This property amounts to linear independence of all such subsets of columns of the parity check matrix.
MR LRCs have received much attention recently, with many explicit constructions covering different regimes of parameters. Unfortunately, all known constructions require a large field size that is exponential in h or a, and it is of interest to obtain MR LRCs of minimal possible field size. In this work, we develop an approach based on function fields to construct MR LRCs. Our method recovers, and in most parameter regimes improves, the field size of previous approaches. For instance, for the case of small r =slant Omega(n^{1-epsilon}), we improve the field size from roughly n^h to n^{epsilon h}. For the case of a=1 (one local parity check), we improve the field size quadratically from r^{h(h+1)} to r^{h floor[(h+1)/2]} for some range of r. The improvements are modest, but more importantly are obtained in a unified manner via a promising new idea
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