Bounds on the Maximal Minimum Distance of Linear Locally Repairable Codes

TL;DR

This paper advances the theoretical understanding of linear locally repairable codes by expanding parameter classes for perfect codes and optimizing bounds on their maximum minimum distance using matroid theory.

Contribution

It extends the existence of perfect linear LRCs to new parameter ranges and refines the lower bounds on maximum minimum distance, proving their optimality within a matroid framework.

Findings

Expanded parameter classes for perfect LRCs.

Improved lower bounds on maximum minimum distance.

Proved bounds are optimal for certain matroid classes.

Abstract

Locally repairable codes (LRCs) are error correcting codes used in distributed data storage. Besides a global level, they enable errors to be corrected locally, reducing the need for communication between storage nodes. There is a close connection between almost affine LRCs and matroid theory which can be utilized to construct good LRCs and derive bounds on their performance. A generalized Singleton bound for linear LRCs with parameters $(n,k,d,r,\delta)$ was given in [N. Prakash et al., "Optimal Linear Codes with a Local-Error-Correction Property", IEEE Int. Symp. Inf. Theory]. In this paper, a LRC achieving this bound is called perfect. Results on the existence and nonexistence of linear perfect $(n,k,d,r,\delta)$-LRCs were given in [W. Song et al., "Optimal locally repairable codes", IEEE J. Sel. Areas Comm.]. Using matroid theory, these existence and nonexistence results were later strengthened in [T. Westerb\"ack et al., "On the Combinatorics of Locally Repairable Codes", Arxiv: 1501.00153], which also provided a general lower bound on the maximal achievable minimum distance $d_{\rm{max}}(n,k,r,\delta)$ that a linear LRC with parameters $(n,k,r,\delta)$ can have. This article expands the class of parameters $(n,k,d,r,\delta)$ for which there exist perfect linear LRCs and improves the lower bound for $d_{\rm{max}}(n,k,r,\delta)$. Further, this bound is proved to be optimal for the class of matroids that is used to derive the existence bounds of linear LRCs.