On the weight hierarchy of locally repairable codes

Abstract

An (n, k, r) 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 ≤ n−k −⌈k/r⌉+2. The 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 ≤ i ≤ k) GHWs of general (n, k, r) LRCs. Then, it is shown that for an optimal (n, k, r) LRC with r | k, its weight hierarchy can be completely determined, and the i-th GHW of an optimal (n, k, r) LRC with r | k attains the proposed generalized Singleton-like bound for all 1 ≤ i ≤ k. For an optimal (n, k, r) LRC with r ∤ 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.

Locally repairable codes with multiple (r<inf>i</inf>, δ<inf>i</inf>)-localities

“…Therefore, defining the largest possible minimum distance of an [n ′ , k ′ ] q linear code by d (opt) q (n ′ , k ′ ), we have the following result [17].…”

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

“…In this paper, we generalize the results of [1] to study the GHWs of this more general class of q-ary (n, k, r, δ)-LRCs with δ ≥ 2 and their dual codes. The upper bounds and lower bounds on GHWs of (n, k, r)-LRCs in [1] can be obtained by the results in this paper when δ = 2. The results of this paper are summarized as follows.…”

“…When r = k, the above bound reduces to the classical Singleton bound d ≤ n − k + 1. An (n, k, r)-LRC is said to be optimal if its minimum distance attains the bound (1).…”

“…When δ = 2, the above bound reduces to the bound (1). Constructions of optimal (n, k, r, δ)-LRCs attaining the bound (2) have also attracted a lot of interests.…”

“…Note that the GHWs of q-ary (n, k, r)-LRCs were studied in [1], which are a special class of (n, k, r, δ)-LRCs with δ = 2. In this paper, we generalize the results of [1] to study the GHWs of this more general class of q-ary (n, k, r, δ)-LRCs with δ ≥ 2 and their dual codes. The upper bounds and lower bounds on GHWs of (n, k, r)-LRCs in [1] can be obtained by the results in this paper when δ = 2.…”