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

The first focus of the present paper, is on lower bounds on the sub-packetization level 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 alpha which is shown to be tight for the case d = (n - 1) by comparing with recent code constructions in the literature. We also extend our results to an n, k] MDS code over the vector alphabet. Our objective even here, is on lower bounds on the sub-packetization level alpha of an MDS code that can carry out repair of any node in a subset of w nodes, 1 <= w <= (n - 1) where each node is repaired (linear repair) by help-by-transfer with minimum repair bandwidth. We prove a lower bound on alpha for the case of d = (n - 1). This bound holds for any w(<= 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 < (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 Q nodes with minimum repair bandwidth and in optimal-access fashion, and which achieve our lower bound on sub-packetization level alpha. It turns out interestingly, that such a code must necessarily have a coupled-layer structure, similar to that of the Ye-Barg code