303 research outputs found
Communication Cost for Updating Linear Functions when Message Updates are Sparse: Connections to Maximally Recoverable Codes
We consider a communication problem in which an update of the source message
needs to be conveyed to one or more distant receivers that are interested in
maintaining specific linear functions of the source message. The setting is one
in which the updates are sparse in nature, and where neither the source nor the
receiver(s) is aware of the exact {\em difference vector}, but only know the
amount of sparsity that is present in the difference-vector. Under this
setting, we are interested in devising linear encoding and decoding schemes
that minimize the communication cost involved. We show that the optimal
solution to this problem is closely related to the notion of maximally
recoverable codes (MRCs), which were originally introduced in the context of
coding for storage systems. In the context of storage, MRCs guarantee optimal
erasure protection when the system is partially constrained to have local
parity relations among the storage nodes. In our problem, we show that optimal
solutions exist if and only if MRCs of certain kind (identified by the desired
linear functions) exist. We consider point-to-point and broadcast versions of
the problem, and identify connections to MRCs under both these settings. For
the point-to-point setting, we show that our linear-encoder based achievable
scheme is optimal even when non-linear encoding is permitted. The theory is
illustrated in the context of updating erasure coded storage nodes. We present
examples based on modern storage codes such as the minimum bandwidth
regenerating codes.Comment: To Appear in IEEE Transactions on Information Theor
Security in Locally Repairable Storage
In this paper we extend the notion of {\em locally repairable} codes to {\em
secret sharing} schemes. The main problem that we consider is to find optimal
ways to distribute shares of a secret among a set of storage-nodes
(participants) such that the content of each node (share) can be recovered by
using contents of only few other nodes, and at the same time the secret can be
reconstructed by only some allowable subsets of nodes. As a special case, an
eavesdropper observing some set of specific nodes (such as less than certain
number of nodes) does not get any information. In other words, we propose to
study a locally repairable distributed storage system that is secure against a
{\em passive eavesdropper} that can observe some subsets of nodes.
We provide a number of results related to such systems including upper-bounds
and achievability results on the number of bits that can be securely stored
with these constraints.Comment: This paper has been accepted for publication in IEEE Transactions of
Information Theor
On Error Decoding of Locally Repairable and Partial MDS Codes
We consider error decoding of locally repairable codes (LRC) and partial MDS
(PMDS) codes through interleaved decoding. For a specific class of LRCs we
investigate the success probability of interleaved decoding. For PMDS codes we
show that there is a wide range of parameters for which interleaved decoding
can increase their decoding radius beyond the minimum distance with the
probability of successful decoding approaching , when the code length goes
to infinity
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