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A Tight Lower Bound on the Sub-Packetization Level of Optimal-Access MSR and MDS Codes

Abstract

The first focus of the present paper, is on lower bounds on the sub-packetization level α\alpha of an MSR code that is capable of carrying out repair in help-by-transfer fashion (also called optimal-access property). We prove here a lower bound on α\alpha which is shown to be tight for the case d=(n1)d=(n-1) by comparing with recent code constructions in the literature. We also extend our results to an [n,k][n,k] MDS code over the vector alphabet. Our objective even here, is on lower bounds on the sub-packetization level α\alpha of an MDS code that can carry out repair of any node in a subset of ww nodes, 1w(n1)1 \leq w \leq (n-1) where each node is repaired (linear repair) by help-by-transfer with minimum repair bandwidth. We prove a lower bound on α\alpha for the case of d=(n1)d=(n-1). This bound holds for any w(n1)w (\leq n-1) and is shown to be tight, again by comparing with recent code constructions in the literature. Also provided, are bounds for the case d<(n1)d<(n-1). We study the form of a vector MDS code having the property that we can repair failed nodes belonging to a fixed set of QQ nodes with minimum repair bandwidth and in optimal-access fashion, and which achieve our lower bound on sub-packetization level α\alpha. It turns out interestingly, that such a code must necessarily have a coupled-layer structure, similar to that of the Ye-Barg code.Comment: Revised for ISIT 2018 submissio

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