Cyclic LRC codes and their subfield subcodes

2017 IEEE International Symposium on Information Theory (ISIT)

2017

In distributed storage systems, locally repairable codes (LRCs) are introduced to realize low disk I/O and repair cost. In order to tolerate multiple node failures, the LRCs with (r, δ)-locality are further proposed. Since hot data is not uncommon in a distributed storage system, both Zeh et al. and Kadhe et al. focus on the LRCs with multiple localities or unequal localities (ML-LRCs) recently, which said that the localities among the code symbols can be different. ML-LRCs are attractive and useful in reducing repair cost for hot data. In this paper, we generalize the ML-LRCs to the (r, δ)-locality case of multiple node failures, and define an LRC with multiple (ri, δi) i∈[s] localities (s ≥ 2), where r1 ≤ r2 ≤ · · · ≤ rs and δ1 ≥ δ2 ≥ · · · ≥ δs ≥ 2. Such codes ensure that some hot data could be repaired more quickly and have better failure-tolerance in certain cases because of relatively smaller ri and larger δi. Then, we derive a Singleton-like upper bound on the minimum distance for the proposed LRCs by employing the regeneratingset technique. Finally, we obtain a class of explicit and structured constructions of optimal ML-LRCs, and further extend them to the cases of multiple (ri, δ) i∈[s] localities.

Section: Remarkmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

2017 IEEE International Symposium on Information Theory (ISIT)

2017

“…Tamo et al [4] proposed an elegant construction of optimal LRCs for q ≥ n + 1 by using polynomial methods. They further proposed optimal cyclic LRCs for q ≥ n + 1 in [5]. Ernvall et al proposed LRCs over a small alphabet in [6].…”

Section: Introductionmentioning

confidence: 99%

On optimal ternary locally repairable codes

2017 IEEE International Symposium on Information Theory (ISIT)

2017

Abstract-In an [n, k, d] linear code, a code symbol is said to have locality r if it can be repaired by accessing at most r other code symbols. For an (n, k, r) locally repairable code (LRC), the minimum distance satisfies the well-known Singleton-like bound d ≤ n − k − ⌈k/r⌉ + 2. In this paper, we study optimal ternary LRCs meeting this Singleton-like bound by employing a paritycheck matrix approach. It is proved that there are only 8 classes of possible parameters with which optimal ternary LRCs exist. Moreover, we obtain explicit constructions of optimal ternary LRCs for all these 8 classes of parameters, where the minimum distance could only be 2, 3, 4, 5 and 6.

“…For an (n, k, r) optimal LRC meeting the Singleton-like bound (1) with r | k, the i-th (1 ≤ i ≤ k) GHW attains the generalized Singleton-like bound (10) with equality. For an (n = 12, k = 6, r = 3) optimal cyclic LRC[7] C with r | k, by Theorem 2, the weight hierarchy of its dual code C ⊥ can be completely determined asd ⊥ 1 = 4, d ⊥ 2 = 8, d ⊥ 3 = 9, d ⊥ 4 = 10, d ⊥ 5 = 11, d ⊥ 6 = 12. By Theorem 3, the complete weight hierarchy of C is d 1 = 6, d 2 = 7, d 3 = 8, d 4 = 10, d 5 = 11, d 6 = 12.…”

mentioning

confidence: 99%

On the weight hierarchy of locally repairable codes

2017 IEEE Information Theory Workshop (ITW)

et al. 2017

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.