Locally Repairable Codes and Matroid Theory
Abstract
Locally repairable codes (LRCs) are error correcting codes used in distributed data storage. A traditional approach is to look for codes which simultaneously maximize error tolerance and minimize storage space consumption. However, this tends to yield codes for which error correction requires an unrealistic amount of communication between storage nodes. LRCs solve this problem by allowing errors to be corrected locally. This thesis reviews previous results on the subject presented in [1]. These include that every almost affine LRC induces a matroid such that the essential properties of the code are determined by the matroid. Also, the generalized Singleton bound for LRCs can be extended to matroids as well. Then, matroid theory can be used to find classes of matroids that either achieve the bound, meaning they are optimal in a certain sense, or at least come close to the bound. This thesis presents an improvement to the results of [1] in both of these cases. [1] T. Westerb\"ack, R. Freij, T. Ernvall and C. Hollanti, "On the Combinatorics of Locally Repairable Codes via Matroid Theory", arXiv:1501.00153 [cs.IT], 2014.
Cite
@article{arxiv.1512.05325,
title = {Locally Repairable Codes and Matroid Theory},
author = {Antti Pöllänen},
journal= {arXiv preprint arXiv:1512.05325},
year = {2015}
}
Comments
Bachelor thesis, Aalto University School of Science, 2015. Instructor Ph.D. Thomas Westerb\"ack, supervisor Prof. Camilla Hollanti. This thesis builds on the article arXiv:1501.00153. Main results: The class of optimal matroids with ceil(k/r)=2 is extended. An improved lower bound for d_max is presented. This bound is proven to be tight for a certain important class of matroids