Linear locally recoverable codes with locality r=1
Samuel Tien Ho Teo · DR-NTU (Nanyang Technological University) · 2017
A code is called a locally recoverable code (LRC) with locality r if any symbol of a codeword can be recovered by accessing r other symbols that forms the recovering set. A LRC has availability t if each symbol has at least t disjoint recovering sets. In this thesis, we summarise the known properties and bounds of linear LRCs and will focus primarily on linear LRCs with locality r = 1 and availability t = 1. We will derive a few propagation rules for linear LRCs with locality r = 1 and present a code construction method using partitions of length n of a LRC. We will prove the optimality of linear LRCs with locality r = 1 for certain values of length n and distance d, and compare upper bounds and lower bounds of binary linear LRCs with locality r = 1 with respect to dimension k. The investigation into the optimal dimensions of linear LRCs is important to improve efficiency in their applications in distributed and cloud storage systems.