On the Generalized Hamming Weights of (r, δ)-Locally Repairable Codes

Jie Hao,Bin Chen

doi:10.1109/access.2020.3016572

Abstract

Locally repairable codes (LRCs) have attracted a lot of interests recently due to their important applications in distributed storage systems. An (n, k, r, δ)-LRC (δ ≥ 2) is an [n, k, d] linear code such that each of the n code symbols satisfies (r, δ)-locality and is said to be optimal if it has minimum distance d = n-k -([k/r]-1)(δ -1)+1. The generalized Hamming weights (GHWs) are fundamental parameters of linear codes. Prakash et al. firstly applied GHWs to study linear codes with locality properties. In this article, we study the GHWs of (n, k, r, δ)-LRCs (δ ≥ 2). Firstly, for a general (n, k, r, δ)-LRC, an upper bound on the i-th (1 ≤ i ≤ k) GHW is presented. Then, for an optimal (n, k, r, δ)-LRC and its dual code, a lower bound on the [i(δ - 1)]-th GHW, for 1 ≤ i ≤ [k/r] - 1, of the dual code is given. Specially, when r k, we determine the [i(δ - 1)]-th GHW, for 1 ≤ i ≤ [k/r] - 1, of the dual code of an optimal (n, k, r, δ)-LRC. For the case of δ = 2, we obtain a lower bound on the i-th GHW for all 1 ≤ i ≤ k of an optimal (n, k, r, 2)-LRC. Moreover, it is shown that the weight hierarchy of an optimal (n, k, r, 2)-LRC with r k can be completely determined.

Highlights

Modern large distributed storage systems usually store redundant data to ensure data reliability in case of storage node failures
For an optimal (n, k, r, δ)-Locally repairable codes (LRCs) and its dual code, a lower bound on the [i(δ − 1)]-th generalized Hamming weights (GHWs), for 1 ≤ i ≤ k/r − 1, of the dual code is given
For q-ary general (n, k, r, δ)-LRCs (δ ≥ 2), an upper bound on the i-th (1 ≤ i ≤ k) GHW is presented, which can be seen as a generalization of the classical generalized Singleton bound by taking the locality constraints into account

Summary

INTRODUCTION

Modern large distributed storage systems usually store redundant data to ensure data reliability in case of storage node failures. Gap numbers were firstly introduced by Prakash et al to derive the bounds on the minimum distance of (r, δ)-LRCs [14] and are useful tools to study GHWs of linear codes. The i-th GHW of an MDS code attains the classical generalized Singleton bound d ≤. LRCs (δ ≥ 2) attaining the Singleton-type bound (2) can be regarded as generalizations of MDS codes with locality constraints. For q-ary general (n, k, r, δ)-LRCs (δ ≥ 2), an upper bound on the i-th (1 ≤ i ≤ k) GHW is presented, which can be seen as a generalization of the classical generalized Singleton bound by taking the locality constraints into account.

THE GHWS OF GENERAL LRCS

THE GHWS OF OPTIMAL LRCS

CONCLUSIONS