基于拟阵理论的局部可修复码的组合性质研究
摘要
本文建立了拟阵理论与线性或更一般地几乎仿射的局部可修复码（LRCs）之间的联系。利用这一联系，导出了关于 LRCs 与拟阵理论的新结果。将 LRCs 的参数 (n,k,d,r,δ) 推广到拟阵，并给出了 [P. Gopalan 等, "On the locality of codeword symbols," IEEE Trans. Inf. Theory] 中线性 LRCs 的广义 Singleton 界的拟阵类比。结果表明，所给界对某些参数类不紧，意味着相应的、本文称为完美的局部可修复几乎仿射码不存在。给出了具有大范围参数 (n,k,d,r,δ) 及其对应局部修复集的拟阵类的构造。利用这些拟阵构造，构建了具有指定参数的新 LRCs。本文给出的线性 LRCs 的存在性结果与几乎仿射 LRCs 的不存在性结果，加强了 [W. Song 等, "Optimal locally repairable codes," IEEE J. Sel. Areas Comm.] 中关于完美线性 LRCs 的不存在性与存在性结果。
引用
@article{arxiv.1501.00153,
title = {On the Combinatorics of Locally Repairable Codes via Matroid Theory},
author = {Thomas Westerbäck and Ragnar Freij and Toni Ernvall and Camilla Hollanti},
journal= {arXiv preprint arXiv:1501.00153},
year = {2016}
}
备注
48 pages. Submitted for publication. In this version: The text has been edited to improve the readability. Parameter d for matroids is now defined by the use of the rank function instead of the dual matroid. Typos are corrected. Section III is divided into two parts, and some numberings of theorems etc. have been changed