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

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


I. INTRODUCTION
Modern large distributed storage systems usually store redundant data to ensure data reliability in case of storage node failures. The redundancy scheme of 3-replication is widely used in distributed storage systems, which stores three replicas of data in different storage nodes. Due to the large volume of data, the 3-replication scheme will introduce large storage overhead. Hence, redundancy schemes based on erasure codes have become more attractive because of their higher storage efficiency compared to replication. Maximum distance separable (MDS) codes, e.g. Reed-Solomon codes, are widely used traditional erasure codes due to their good ability of fault tolerance. For storage systems using Reed-Solomon codes, the data is firstly partitioned into k packets, then the [n, k] Reed-Solomon code encodes k packets into n packets and stores them across n storage nodes. The MDS property ensure the data reliability in case of any n − k failures. However, when some storage node fails, the storage systems need to repair the failed storage node. The repair process requires The associate editor coordinating the review of this manuscript and approving it for publication was Keivan Navaie . reading packets from k other surviving nodes, which introduces large amount of repair bandwidth. Locally repairable codes (LRCs) [2] are a class of improved erasure codes which can repair failed storage nodes efficiently and have attracted a lot of interests recently. Several real distributed storage systems, e.g., Microsoft Azure Storage [3] and Hadoop Distributed File System [4], have deployed LRCs as their redundancy schemes. Another important class of storage codes which can also achieve efficient repairing of failed storage nodes are regenerating codes [5]. Some works have proposed codes to combine the repair benefits of locally repairable codes and regenerating codes. Codes with local regeneration and erasure correction were proposed in [6]. Repair duality with locally repairable and locally regenerating codes were studied in [7].
In this article, we focus on locally repairable codes. In a q-ary [n, k, d] linear code, a code symbol is said to have r-locality if it can be repaired by accessing at most r other code symbols. A q-ary (n, k, r)-LRC is a q-ary [n, k, d] linear code with r-locality for all code symbols. When r k, the storage system using LRCs as redundancy scheme only needs to read data from a small number of other available nodes to repair a failed storage node, which indicates low repair bandwidth. Gopalan et al. proved that the minimum distance of a q-ary (n, k, r)-LRC satisfies the following well-known Singleton-type bound [2], 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). The elegant Reed-Solomon-like optimal LRCs [8] require the field size to be just q ≥ n. Classifications of optimal binary and ternary (n, k, r)-LRCs meeting the bound (1) were given in [9], [10]. Optimal cyclic (n, k, r)-LRCs with code length up to q − 1 were given in [11]. Luo et al. proposed q-ary optimal cyclic (n, k, r)-LRCs with d = 3, 4 and unlimited length [12]. Wang and Zhang proposed a refined bound of (n, k, r)-LRCs based on integer programming methods [13].
In order to ensure local recovery of a failed node in case of more than one node failures, Prakash et al. firstly proposed the concept of (r, δ)-locality [14].
• the minimum distance of the punctured subcode C| i , obtained by puncturing these code symbols in {c j , j ∈ [n] \ i }, is at least δ. From the view of parity-check matrices, suppose that the punctured local subcode C| i has dimension k i ≤ r, then there exist | i | − k i linearly independent codewords from the dual code C ⊥ which form a local parity-check matrix, or local-PCM for short, H i of the local subcode C| i such that the following three conditions are satisfied.
• any δ − 1 columns of these |supp(H i )| columns in H i are linearly independent. The set supp(H i ) denotes the union of the supports of all the codewords in H i and |supp(H i )| is the cardinality of the set supp(H i ). The code symbol c i (or the coordinate i) is said to be covered by the local-PCM H i . The local-PCM H i contains at least δ − 1 linearly independent codewords. An (n, k, r, δ)-LRC (δ ≥ 2) is an [n, k, d] linear code with (r, δ)-locality for all code symbols, i.e., each code symbol is covered by a local-PCM satisfying the above three conditions. The minimum distance of an (n, k, r, δ)-LRC satisfies the following Singleton-type bound [14], 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. Optimal q-ary cyclic (n, k, r, δ)-LRCs with code length q + 1 were proposed in [15]. Classifications of optimal binary and ternary (n, k, r, δ)-LRCs with δ > 2 were given in [16] and [17], respectively. Constructions of optimal (n, k, r, δ)-LRCs over a small alphabet were proposed in [18]. Other optimal constructions can be found in [14], [19]- [22]. Note that there is another type of generalization of r-locality called (r, t)-locality in which each code symbol has t disjoint groups of other code symbols to repair it, each group of size at most r. Bounds and constructions of such LRCs with multiple disjoint repair groups can be found in [23]- [28].
The generalized Hamming weights (GHWs) [29], [30] are fundamental parameters of linear codes, which were first used by Wei to characterize the performance of linear codes in a wire-tap channel of type II [29]. Let D be a subcode of an [n, k, d] linear code C. Denote the dimension of D as dim (D). The support of the subcode D is defined to be When i = 1, the first GHW d 1 is exactly the minimum distance of C. The i-th GHW, for 1 ≤ i ≤ k, of C satisfies the following generalized Singleton bound, When i = 1, the bound (3) gives the classical Singleton bound. The weight hierarchy of C is the set of GHWs which is the complement of its weight hierarchy. 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 weight hierarchy and gap numbers of the Many works have conducted research to determine or estimate the GHWs of different linear codes, such as Hamming codes [29], Reed-Muller codes [29], [31], BCH codes and their dual codes [32]- [34], etc.. Generally speaking, it is difficult to determine the GHWs of a linear code, especially for the complete weight hierarchy, which is known for only a few cases of linear codes. Prakash et al. firstly introduced GHWs to study linear codes with locality properties [14] and used GHWs to study codes with a sequential repair property for two erasures [35]. These two papers are pioneer work which introduced GHWs to study LRCs. Ballico and Marcolla [36] studied the GHWs of LRCs on algebraic curves. Lalitha and Lokam [37] studied the GHWs of maximally recoverable codes [38]. Some techniques in [14] can also be used to determine the weight hierarchy of maximally recoverable codes.
It is well known that for an [n, k, d] MDS code attaining the classical Singleton bound, its weight hierarchy can be VOLUME 8, 2020 uniquely determined. Moreover, the i-th GHW of an MDS code attains the classical generalized Singleton bound d ≤ n − k + i for all 1 ≤ i ≤ k. The Singleton-type bound (1) and (2) are generalizations of the classical Singleton bound by taking the locality property into account. Optimal (n, k, r, δ)-LRCs (δ ≥ 2) attaining the Singleton-type bound (2) can be regarded as generalizations of MDS codes with locality constraints. It is an interesting problem to investigate the properties of GHWs of q-ary (n, k, r, δ)-LRCs (δ ≥ 2).
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 article, 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 article when δ = 2. The results of this article are summarized as follows.
• Firstly, 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. Particularly, when i = 1, the proposed upper bound on GHWs gives the Singleton-type upper bound (2) on the minimum distance of (r, δ)-LRCs. When r = k, the proposed upper bound reduces to the classical generalized Singleton bound (3).
• Then, we focus on the GHWs of q-ary optimal (n, k, r, δ)-LRCs (δ ≥ 2) and their dual codes. A lower bound on the [i(δ − 1)]-th GHW, for 1 ≤ i ≤ k r − 1, of the dual code of a q-ary optimal (n, k, r, δ)-LRC is given. When the dimension k is divisible by r, we determine the [i(δ −1)]-th GHWs, 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 present a lower bound on the i-th GHW, for all 1 ≤ i ≤ k, of a q-ary optimal (n, k, r, 2)-LRC. Moreover, it is shown that the weight hierarchy of a q-ary optimal (n, k, r, 2)-LRC with r | k can be completely determined.
The rest of this article is organized as follows. Section II presents an upper bound on the GHWs of q-ary general (n, k, r, δ)-LRCs (δ ≥ 2). In Section III, we focus on the GHWs of optimal (n, k, r, δ)-LRCs and their dual codes. Section IV concludes the paper.

II. THE GHWs OF GENERAL LRCs
In this section, we consider the GHWs of q-ary general (n, k, r, δ)-LRCs (δ ≥ 2). An upper bound on the i-th GHW, for 1 ≤ i ≤ k, of q-ary general (n, k, r, δ)-LRCs is presented, which can include the Singleton-type bound (2) and the generalized Singleton bound (3) as special cases.
Lemma 1: Let C be a q-ary (n, k, r, δ)-LRC (δ ≥ 2) and C ⊥ be its dual code. Then, the i-th GHW, for Since all the n code symbols of C satisfy (r, δ)-locality, each of the n coordinates is contained in a punctured local subcode with length at most r + δ − 1 and minimum distance at least δ. From the view of parity-check matrices, each of the n coordinates is covered by a local-PCM which has at least δ − 1 linearly independent codewords from the dual code C ⊥ and the support of such local-PCM has size at most r + δ − 1. In the following, for each 1 ≤ i ≤ ( k/r −1)(δ−1), we select i linearly independent codewords from some i δ−1 = i−1 δ−1 + 1 local-PCMs such that these i linearly independent codewords form a group of basis of an i-dimensional subcode D ⊥ i of the dual code C ⊥ . Write where 1 ≤ φ ≤ δ − 1. The i selected linearly independent codewords will contain i−1 δ−1 (δ − 1) codewords from some i−1 δ−1 local-PCMs and additional φ codewords from one local-PCM. The details of the selection procedures are as follows.
• Step 1: Firstly, for the first coordinate, it is covered by a local-PCM, denoted by H 1 , which contains ≥ δ − 1 linearly independent codewords from the dual code C ⊥ . Select δ − 1 linearly independent codewords from this local-PCM H 1 , by the property of (r, δ)-locality, these δ −1 selected codewords from H 1 cover at most r +δ −1 coordinates.
• Step 2: Then, choose a coordinate outside the union of supports of the previous selected codewords such that this coordinate satisfies the condition that 'the coordinate is covered by a local-PCM H * in which at least δ − 1 codewords are linearly independent with all the previous selected codewords'. Select δ −1 linearly independent codewords from this local-PCM H * so that all the selected codewords are linearly independent.
• Step 3: Repeat the above procedure in Step 2 iteratively to select i−1 δ−1 (δ − 1) codewords from some i−1 δ−1 local-PCMs in total, such that all these i−1 δ−1 (δ − 1) selected codewords are linearly independent and the support of each group of δ − 1 codewords from one local-PCM cover at most r + δ − 1 coordinates. • Step 4: Choose a coordinate outside the union of supports of the previous selected codewords which satisfies the condition in Step 2. Denote H as the δ−1 codewords in the local-PCM of this coordinate which are linearly independent with all the previous selected codewords. We transform the right δ − 1 columns of H into an identity matrix by linear combinations of these δ − 1 rows. Then select the first φ rows. Note that by [16,Lemma 1], when the number of the selected local-PCMs is less than k/r −1, a coordinate satisfying the condition in Step 2 always exist. For the complexity of the selection procedure, since 1 ≤ i ≤ ( k r − 1)(δ − 1), the above selection procedure only needs to select codewords from i δ−1 ≤ k r − 1 local-PCMs. Finally, a total of i−1 δ−1 (δ − 1) + φ = i linearly independent codewords are selected from the dual code C ⊥ . The following figure illustrates the result of the above procedure briefly.
By the above selection procedure, all these i selected codewords are linearly independent. Hence, they form a group of basis local-PCMs cover at most i−1 δ−1 (r + δ − 1) coordinates, and the last φ codewords cover at most r + φ coordinates. Then, the support of this i-dimensional subcode D ⊥ i satisfies this completes the proof. Lemma 2: Let C be a q-ary (n, k, r, δ)-LRC (δ ≥ 2) and C ⊥ be its dual code. When Let H i be a group of basis of the subcode D ⊥ i which consists of i linearly independent codewords from C ⊥ and where the right δ−1 columns of H * δ−1 is an identity matrix. By the property of (r, δ)-locality, |supp( H * δ−1 )| ≤ r + δ − 1, thus the left part H δ−1 of H * δ−1 has support size |supp( H δ−1 )| ≤ r. Hence, the first t rows, for 1 ≤ t ≤ δ − 1, of H * δ−1 cover at most r + t coordinates. Denote the first t rows of H * δ−1 as H * t . Then |supp(H * t )| ≤ r + t. Let Since all the t rows of H * t are linearly independent with all the i rows in H i , it follows that H i+t has full rank. Therefore, the i+t codewords in H i+t form a group of basis of a subcode Then, we obtain that the (i + t)-th GHW, for Hence, the conclusion holds.
Lemma 3: Let C be a q-ary (n, k, r, δ)-LRC (δ ≥ 2) and C ⊥ be its dual code. Then the i-th (1 ≤ i ≤ k) gap number of the dual code C ⊥ satisfies Proof: We prove the result of this lemma by induction. Firstly, when i = 1, the first gap number of C ⊥ satisfies g ⊥ 1 = 1. Otherwise, if g ⊥ 1 ≥ 2, then d ⊥ 1 = 1. In this case, there exist at least one codeword in the dual code C ⊥ which has weight one. This indicates that there exist at least one specific coordinate of C at which the corresponding code symbol of all codewords of C always equals zero.
Next, assume that for 1 ≤ i ≤ k − 1, the i-th gap number 2) If r | i, then i+1 r = i r + 1. Let i = sr for some integer s ≤ k−1 r = k r − 1. Then, i + 1 = sr + 1. By Lemma 1, the [s(δ − 1)]-th GHW of the dual code . VOLUME 8, 2020 Hence, the set {1, 2, · · · , s(r + δ − 1)} contains at least the GHWs of C ⊥ in the following set On the other hand, this indicates that the set {1, 2, · · · , s(r + δ − 1)} contains at most these gap numbers of C ⊥ in the following set {g ⊥ 1 , g ⊥ 2 , · · · , g ⊥ sr }. Therefore, the (sr + 1)-th gap number of the dual code Combining all the above discussions, the lemma holds. Theorem 1: Let C be a q-ary (n, k, r, δ)-LRC (δ ≥ 2), then the i-th GHW, for 1 ≤ i ≤ k, of C satisfies (4), we know that the i-th GHW, for 1 ≤ i ≤ k, of C satisfies By Lemma 3, it follows that Combining (12) and (13), we have this completes the proof. Remark 1: The upper bound (11) on the GHWs of (n, k, r, δ)-LRCs can be regarded as a generalization of classical generalized Singleton bound (3) by taking the (r, δ)-locality properties of code symbols into account.
• When r = k, i.e., the locality properties of code symbols are not considered, the upper bound (11) reduces to the classical generalized Singleton bound (3). By Theorem 1, when δ = 2, the bound (11) gives the following upper bound of i-th GHW, for 1 ≤ i ≤ k, of q-ary (n, k, r)-LRCs.
Corollary 1: Let C be a q-ary (n, k, r)-LRC, then the i-th GHW, for 1 ≤ i ≤ k, of C satisfies Remark 2: Note that there is another different kind of locality, called (r, e)-cooperative locality [39], which can also enable local recovery of failed storage nodes in case of at most e node failures. When e = 1, the concept of (r, e)-cooperative locality reduces to r-locality. Abdel-Ghaffar and Weber [40] presented an upper bound on the GHWs of q-ary [n, k, d] linear codes with (r, e)-cooperative locality [40,Theorem 3]. When δ = 2 and e = 1, both our bound (11) in Theorem 1 and the bound in [40,Theorem 3] reduce to the bound on the GHWs of LRCs with r-locality in Corollary 1.

III. THE GHWs OF OPTIMAL LRCs
In this section, we consider the GHWs of q-ary optimal (n, k, r, δ)-LRCs (δ ≥ 2) and their dual codes. Firstly, a lower bound on the [i(δ −1)]-th (1 ≤ i ≤ k r −1) GHW of the dual code of a q-ary optimal (n, k, r, δ)-LRC is given. Specially, when r | k, we exactly determine the [i(δ − 1)]-th GHW, for 1 ≤ i ≤ k r −1, of the dual code. Then, for the case of δ = 2, a lower bound on the i-th (1 ≤ i ≤ k) GHW of a q-ary optimal (n, k, r, 2)-LRC is presented. Furthermore, it is shown that the weight hierarchy of a q-ary optimal (n, k, r, 2)-LRC with r | k can be completely determined.
Theorem 4: Let C be a q-ary optimal (n, k, r, 2)-LRC attaining the Singleton-type bound (2), then the i-th GHW, for 1 ≤ i ≤ k, of C satisfies Proof: For 1 ≤ i ≤ k, by (4), it follows that the i-th GHW of C satisfies VOLUME 8, 2020 By Lemma 4, it follows that Combining (25) and (26), we have Hence, the conclusion holds. Generally, the weight hierarchy of q-ary optimal (n, k, r, 2)-LRCs attaining the Singleton-type bound (2) can not be uniquely determined. In the following, we show that for a q-ary optimal (n, k, r, 2)-LRC with dimension k divisible by r, its weight hierarchy can be uniquely determined.
Theorem 5: Let C be a q-ary optimal (n, k, r, 2)-LRC attaining the Singleton-type bound (2) with dimension k divisible by r. Then the i-th GHW, for 1 ≤ i ≤ k, satisfies Proof: By the upper bound (14) in Theorem 1, we know when δ = 2, the i-th (1 ≤ i ≤ k) GHW of C satisfies Meanwhile, by the lower bound (24) of GHWs of q-ary optimal (n, k, r, 2)-LRC in Theorem 4, together with the condition that r | k, we know the i-th (1 ≤ i ≤ k) GHW of C satisfies Combining (28) and (29), when r | k, the i-th GHW, for 1 ≤ i ≤ k, of C is this completes the proof. Example 2: For an optimal cyclic (n = 16, k = 8, r = 4)-LRC [11] C over F 17 with d = 8 and r | k, by Theorem 5, the weight hierarchy of C can be completely determined as follows, It can be easily verified that i-th GHW of this optimal cyclic (16,8,4)-LRC attains the generalized Singleton-like bound (14) for all 1 ≤ i ≤ 8.

IV. CONCLUSION
In this article, we study the generalized Hamming weights of q-ary (n, k, r, δ)-LRCs (δ ≥ 2) and their dual codes. Firstly, for q-ary general (n, k, r, δ)-LRCs, an upper bound on the i-th GHW, for 1 ≤ i ≤ k, is presented. When i = 1, the proposed upper bound gives the Singleton-type bound (2). When r = k, it reduces to the classical generalized Singleton bound (3). Then, for a q-ary optimal (n, k, r, δ)-LRC attaining the Singleton-type bound (2), a lower bound on the [i(δ − 1)]-th GHW, for 1 ≤ i ≤ k/r − 1, of the dual code is given. When the dimension k is divisible by r, we show that the [i(δ − 1)]-th GHW, for 1 ≤ i ≤ k/r − 1, of the dual code of a q-ary optimal (n, k, r, δ)-LRC can be exactly determined. For the case of δ = 2, a lower bound on the i-th (1 ≤ i ≤ k) GHW of a q-ary optimal (n, k, r, 2)-LRC is obtained. Moreover, it is shown that the weight hierarchy of a q-ary optimal (n, k, r, 2)-LRC with r | k can be completely determined.

ACKNOWLEDGMENT
This article was presented in part at the 2017 IEEE Information Theory Workshop.