On the Weight Hierarchy of Locally Repairable Codes

TL;DR

This paper investigates the generalized Hamming weights of locally repairable codes (LRCs), establishing bounds and exact values for optimal codes, thereby advancing understanding of their weight hierarchy and related parameters.

Contribution

It derives a generalized Singleton-like bound on GHWs of LRCs and completely determines the weight hierarchy for optimal codes with divisible r, also providing bounds for non-divisible cases.

Findings

Established a generalized Singleton-like bound on GHWs of LRCs.

Determined the weight hierarchy for optimal (n,k,r) LRCs with r dividing k.

Provided lower bounds on GHWs for non-divisible r cases and bounds for dual codes.

Abstract

An $(n,k,r)$ \emph{locally repairable code} (LRC) is an $[n,k,d]$ linear code where every code symbol can be repaired from at most $r$ other code symbols. An LRC is said to be optimal if the minimum distance attains the Singleton-like bound $d \le n-k-\lceil k/r \rceil +2$. The \emph{generalized Hamming weights} (GHWs) of linear codes are fundamental parameters which have many useful applications. Generally it is difficult to determine the GHWs of linear codes. In this paper, we study the GHWs of LRCs. Firstly, we obtain a generalized Singleton-like bound on the $i$-th $(1\le i \le k)$ GHWs of general $(n,k,r)$ LRCs. Then, it is shown that for an optimal $(n,k,r)$ LRC with $r \mid k$, its weight hierarchy can be completely determined, and the $i$-th GHW of an optimal $(n,k,r)$ LRC with $r \mid k$ attains the proposed generalized Singleton-like bound for all $1\le i \le k$. For an optimal $(n,k,r)$ LRC with $r \nmid k$, we give lower bounds on the GHWs of the LRC and its dual code. Finally, two general bounds on linear codes in terms of the GHWs are presented. Moreover, it is also shown that some previous results on the bounds of minimum distances of linear codes can also be explained or refined in terms of the GHWs.