Optimal Locally Repairable Constacyclic Codes of Prime Power Lengths - - PowerPoint PPT Presentation
Optimal Locally Repairable Constacyclic Codes of Prime Power Lengths Wei Zhao 1 , 2 , Kenneth W. Shum 1 and Shenghao Yang 1 1 The Chinese University of Hong Kong, Shenzhen 2 University of Science and Technology of China ISIT 2020 Zhao, Shum, Yang
Wei Zhao1,2, Kenneth W. Shum1 and Shenghao Yang1
1The Chinese University of Hong Kong, Shenzhen 2University of Science and Technology of China
ISIT 2020
Zhao, Shum, Yang (CUHK(SZ) & USTC) Optimal LRCs ISIT 2020 1 / 16
Reliability to access data
◮ replication: large storage overhead ◮ erasure coding: relatively smaller storage overhead
Storage systems deploying coding techniques
◮ Window Azure [Huang, Simitci, and et al., 2012] ◮ Facebook analytics Hadoop cluster [Sathiamoorthy, Asteris, and et al.,
2013]
Node repair is important to maintain the failure tolerance capability
Three major repair cost metrics for node repair
◮ the repair bandwidth [Dimakis, Godfrey, and et al., 2010] ◮ the disk I/O [Khan, Burns, Plank, and Huang, 2011] ◮ the repair locality [Huang, Chen, and Li, 2007] Zhao, Shum, Yang (CUHK(SZ) & USTC) Optimal LRCs ISIT 2020 2 / 16
Reliability to access data
◮ replication: large storage overhead ◮ erasure coding: relatively smaller storage overhead
Storage systems deploying coding techniques
◮ Window Azure [Huang, Simitci, and et al., 2012] ◮ Facebook analytics Hadoop cluster [Sathiamoorthy, Asteris, and et al.,
2013]
Node repair is important to maintain the failure tolerance capability
Three major repair cost metrics for node repair
◮ the repair bandwidth [Dimakis, Godfrey, and et al., 2010] ◮ the disk I/O [Khan, Burns, Plank, and Huang, 2011] ◮ the repair locality [Huang, Chen, and Li, 2007] Zhao, Shum, Yang (CUHK(SZ) & USTC) Optimal LRCs ISIT 2020 2 / 16
Reliability to access data
◮ replication: large storage overhead ◮ erasure coding: relatively smaller storage overhead
Storage systems deploying coding techniques
◮ Window Azure [Huang, Simitci, and et al., 2012] ◮ Facebook analytics Hadoop cluster [Sathiamoorthy, Asteris, and et al.,
2013]
Node repair is important to maintain the failure tolerance capability
Three major repair cost metrics for node repair
◮ the repair bandwidth [Dimakis, Godfrey, and et al., 2010] ◮ the disk I/O [Khan, Burns, Plank, and Huang, 2011] ◮ the repair locality [Huang, Chen, and Li, 2007] Zhao, Shum, Yang (CUHK(SZ) & USTC) Optimal LRCs ISIT 2020 2 / 16
Reliability to access data
◮ replication: large storage overhead ◮ erasure coding: relatively smaller storage overhead
Storage systems deploying coding techniques
◮ Window Azure [Huang, Simitci, and et al., 2012] ◮ Facebook analytics Hadoop cluster [Sathiamoorthy, Asteris, and et al.,
2013]
Node repair is important to maintain the failure tolerance capability
Three major repair cost metrics for node repair
◮ the repair bandwidth [Dimakis, Godfrey, and et al., 2010] ◮ the disk I/O [Khan, Burns, Plank, and Huang, 2011] ◮ the repair locality [Huang, Chen, and Li, 2007] Zhao, Shum, Yang (CUHK(SZ) & USTC) Optimal LRCs ISIT 2020 2 / 16
Given a block code C of length n over Fq, a position i has locality r if there exists λjt ∈ Fq, where 1 ≤ t ≤ r and jt = i, such that for each codeword (c0, c1, · · · , cn−1) ∈ C ci =
r
λjtcjt (1) C has locality r if all the positions have locality r. Codes with a low locality can benefit the reconstruction of failed nodes, also called LRCs.
Zhao, Shum, Yang (CUHK(SZ) & USTC) Optimal LRCs ISIT 2020 3 / 16
Given a block code C of length n over Fq, a position i has locality r if there exists λjt ∈ Fq, where 1 ≤ t ≤ r and jt = i, such that for each codeword (c0, c1, · · · , cn−1) ∈ C ci =
r
λjtcjt (1) C has locality r if all the positions have locality r. Codes with a low locality can benefit the reconstruction of failed nodes, also called LRCs.
Zhao, Shum, Yang (CUHK(SZ) & USTC) Optimal LRCs ISIT 2020 3 / 16
Given a block code C of length n over Fq, a position i has locality r if there exists λjt ∈ Fq, where 1 ≤ t ≤ r and jt = i, such that for each codeword (c0, c1, · · · , cn−1) ∈ C ci =
r
λjtcjt (1) C has locality r if all the positions have locality r. Codes with a low locality can benefit the reconstruction of failed nodes, also called LRCs.
Zhao, Shum, Yang (CUHK(SZ) & USTC) Optimal LRCs ISIT 2020 3 / 16
The Singleton-like bound for an [n, k, d] code with locality r is n − k ≥ k r
(2) A tradeoff between the locality and the ability to correct erasures. LRCs achieve the bound with equality are said to be optimal.
Zhao, Shum, Yang (CUHK(SZ) & USTC) Optimal LRCs ISIT 2020 4 / 16
For a unit λ ∈ Fq, the λ-constacyclic shift τλ on Fn
q is
τλ(c0, c1, . . . , cn−1) = (λcn−1, c0, . . . , cn−2). (3) A linear code C is said to be λ-constacyclic if τλ(C) = C. C is a cyclic code if λ = 1. Polynomial representation: c = (c0, c1, . . . , cn−1) ↔ c(x) = c0 + c1x + · · · + cn−1xn−1 (4) τλ(c) ↔ xc(x) ∈ Fq[x]/xn − λ (5) A λ-constacyclic code ↔ An ideal of Fq[x]/xn − λ (6)
Zhao, Shum, Yang (CUHK(SZ) & USTC) Optimal LRCs ISIT 2020 5 / 16
For a unit λ ∈ Fq, the λ-constacyclic shift τλ on Fn
q is
τλ(c0, c1, . . . , cn−1) = (λcn−1, c0, . . . , cn−2). (3) A linear code C is said to be λ-constacyclic if τλ(C) = C. C is a cyclic code if λ = 1. Polynomial representation: c = (c0, c1, . . . , cn−1) ↔ c(x) = c0 + c1x + · · · + cn−1xn−1 (4) τλ(c) ↔ xc(x) ∈ Fq[x]/xn − λ (5) A λ-constacyclic code ↔ An ideal of Fq[x]/xn − λ (6)
Zhao, Shum, Yang (CUHK(SZ) & USTC) Optimal LRCs ISIT 2020 5 / 16
For a λ-constacyclic code C, there exists a unique polynomial g(x) =
(x − αi) (7) to generate C (called the generator polynomial of C), where α is a primitive n-th root of unity and T is a subset of {0, 1, · · · , n − 1}. Z = {αi | i ∈ T} is called the zeros of C.
Zhao, Shum, Yang (CUHK(SZ) & USTC) Optimal LRCs ISIT 2020 6 / 16
Most existing constructions of optimal cyclic LRCs use zero structures:
1 If Z ⊃ {αi(r+1) | 0 ≤ i ≤
n r+1 − 1}, then C has locality r.
2 If Z ⊃ {αb, αb+1, · · · , αb+δ−2}, then the minimum distance of C is at
least δ (BCH bound).
3 Designing the value of b and δ such that n − k = ⌈ k
r ⌉ + δ − 2.
Zhao, Shum, Yang (CUHK(SZ) & USTC) Optimal LRCs ISIT 2020 7 / 16
As far as we know, the constructions of optimal cyclic LRCs in the literatures have a constraint on the locality that r + 1 must divides n due to the using of zero structures. If r + 1 does not divide n, then how to construct the optimal cyclic LRCs? For now, there are no constructions of binary or ternary optimal cyclic LRCs with unbounded code length, while the binary or ternary case are also of interest in theory and practice.
Zhao, Shum, Yang (CUHK(SZ) & USTC) Optimal LRCs ISIT 2020 8 / 16
As far as we know, the constructions of optimal cyclic LRCs in the literatures have a constraint on the locality that r + 1 must divides n due to the using of zero structures. If r + 1 does not divide n, then how to construct the optimal cyclic LRCs? For now, there are no constructions of binary or ternary optimal cyclic LRCs with unbounded code length, while the binary or ternary case are also of interest in theory and practice.
Zhao, Shum, Yang (CUHK(SZ) & USTC) Optimal LRCs ISIT 2020 8 / 16
We mainly consider the λ-constacyclic codes of length ps over finite field Fpm, which are the ideals of Fpm[x]/xps − λ. Use (n, k, d, r)q to denote a q-ary [n, k, d] LRC with locality r. We show that all the optimal constacyclic LRCs of prime power lengths over finite field fall into seven classes: C.1 (2s, 2s−1 − 1, 4, 1)2m, where s ≥ 2; C.2 (ps, ps − ps−k−1, 2, pk+1 − 1)pm, where 1 ≤ k ≤ s − 1 and s ≥ 2; C.3 (ps, ps − 2, 2, ps − ps−1 − 1)pm, where p ≥ 3 and s ≥ 2; C.4 (ps, ps − ps−1 − 1, 3, p − 1)pm, where p ≥ 3 and s ≥ 2; C.5 (ps, 1, ps, 1)pm, where s ≥ 2; C.6 (ps, ps − ps−1, 2, p − 1)pm; C.7 (p, p − t − 1, t + 2, p − t − 1)pm, where 1 ≤ t ≤ p − 2 and p ≥ 3.
Zhao, Shum, Yang (CUHK(SZ) & USTC) Optimal LRCs ISIT 2020 9 / 16
The optimal LRCs in Class 1-6 are of unbounded lengths, where Class 1, 2, 5 and 6 contain binary and ternary LRCs, and Class 3 and 4 contain ternary LRCs. For any integer s ≥ 2, our constructions contain the following two
◮ A binary (2s, 2s−1 − 1, 4, 1) optimal LRC; ◮ A ternary (3s, 3s − 3s−1 − 1, 3, 2) optimal LRC.
Class 3, Class 5 and Class 7 show the optimal constacyclic LRCs where r + 1 does not divide n.
◮ A ternary (3s, 3s − 2, 2, 2 · 3s−1 − 1) optimal LRC; ◮ A 5-ary (5, 3, 3, 3) optimal LRC.
Class 1 includes a class of binary repeated-root cyclic codes, which have many desirable properties [Massey, Costello, and Justesen, 1973].
Zhao, Shum, Yang (CUHK(SZ) & USTC) Optimal LRCs ISIT 2020 10 / 16
The optimal LRCs in Class 1-6 are of unbounded lengths, where Class 1, 2, 5 and 6 contain binary and ternary LRCs, and Class 3 and 4 contain ternary LRCs. For any integer s ≥ 2, our constructions contain the following two
◮ A binary (2s, 2s−1 − 1, 4, 1) optimal LRC; ◮ A ternary (3s, 3s − 3s−1 − 1, 3, 2) optimal LRC.
Class 3, Class 5 and Class 7 show the optimal constacyclic LRCs where r + 1 does not divide n.
◮ A ternary (3s, 3s − 2, 2, 2 · 3s−1 − 1) optimal LRC; ◮ A 5-ary (5, 3, 3, 3) optimal LRC.
Class 1 includes a class of binary repeated-root cyclic codes, which have many desirable properties [Massey, Costello, and Justesen, 1973].
Zhao, Shum, Yang (CUHK(SZ) & USTC) Optimal LRCs ISIT 2020 10 / 16
The optimal LRCs in Class 1-6 are of unbounded lengths, where Class 1, 2, 5 and 6 contain binary and ternary LRCs, and Class 3 and 4 contain ternary LRCs. For any integer s ≥ 2, our constructions contain the following two
◮ A binary (2s, 2s−1 − 1, 4, 1) optimal LRC; ◮ A ternary (3s, 3s − 3s−1 − 1, 3, 2) optimal LRC.
Class 3, Class 5 and Class 7 show the optimal constacyclic LRCs where r + 1 does not divide n.
◮ A ternary (3s, 3s − 2, 2, 2 · 3s−1 − 1) optimal LRC; ◮ A 5-ary (5, 3, 3, 3) optimal LRC.
Class 1 includes a class of binary repeated-root cyclic codes, which have many desirable properties [Massey, Costello, and Justesen, 1973].
Zhao, Shum, Yang (CUHK(SZ) & USTC) Optimal LRCs ISIT 2020 10 / 16
The locality can be characterized by the dual distance of the code. For a code C, if there exits c = (c0, c1, · · · , cn−1) ∈ C⊥ such that w(c) = r + 1 and support(c) = {i0, i1, · · · , ir}, then the positions ij, where 0 ≤ j ≤ r, has locality r. For a constacyclic code, the locality equals to the dual distance minus
Zhao, Shum, Yang (CUHK(SZ) & USTC) Optimal LRCs ISIT 2020 11 / 16
The locality can be characterized by the dual distance of the code. For a code C, if there exits c = (c0, c1, · · · , cn−1) ∈ C⊥ such that w(c) = r + 1 and support(c) = {i0, i1, · · · , ir}, then the positions ij, where 0 ≤ j ≤ r, has locality r. For a constacyclic code, the locality equals to the dual distance minus
Zhao, Shum, Yang (CUHK(SZ) & USTC) Optimal LRCs ISIT 2020 11 / 16
Consider the constacyclic code C = (x − λ0)ps−1. One can verify that the code length of C is ps and the dimension of C is ps − ps−1. Define Ct = gt(x) with gt(x) = x − λ0, if 0 ≤ t ≤ ps−1 − 1 1, if ps−1 ≤ t ≤ ps − 1 (8) Let [tm−1, · · · , t1, t0] be the radix-p expansion of t, Vt =
m−1
(tj + 1). (9)
Zhao, Shum, Yang (CUHK(SZ) & USTC) Optimal LRCs ISIT 2020 12 / 16
The minimum distance of C is d(C) = min{Vt · d(Ct)|0 ≤ t ≤ ps − 1}. (10) If 0 ≤ t ≤ ps−1 − 1, then Ct = 0 and by convention d(Ct) = ∞. If ps−1 ≤ t ≤ ps − 1, then Ct = 1 and hence d(Ct) = 1. By (10), d(C) = Vps−1 · 1 = 2. The dual code of C has the same parameters with code Cps−ps−1. By the same method, d⊥ = p. The locality of C is d⊥ − 1 = p − 1. The [ps, ps − ps−1, 2] code C is an optimal LRC with locality p − 1.
Zhao, Shum, Yang (CUHK(SZ) & USTC) Optimal LRCs ISIT 2020 13 / 16
In this paper, we found all the optimal LRCs within the class of constacyclic codes of prime power lengths, which have the desired small locality and long code length. We present the binary and ternary optimal constacyclic LRCs with unbounded code length. Besides, the explicit constructions of optimal constacyclic LRCs with r plus one does not divide the code length are obtained. Search out the optimal constacyclic LRCs of length ηps over Fq, where η is a positive integer coprime to p, in order to find out the
In the future work, we will characterize the (r, δ)-locality of constacyclic codes and construct optimal (r, δ)-LRCs with low locality and large distance using constacyclic codes.
Zhao, Shum, Yang (CUHK(SZ) & USTC) Optimal LRCs ISIT 2020 14 / 16
Zhao, Shum, Yang (CUHK(SZ) & USTC) Optimal LRCs ISIT 2020 15 / 16
Alexandros Dimakis, P Godfrey, and et al. Network coding for distributed storage
Cheng Huang, Minghua Chen, and Jin Li. Pyramid codes: Flexible schemes to trade space for access efficiency in reliable data storage systems. In Sixth IEEE International Symposium on Network Computing and Applications (NCA 2007), 12 - 14 July 2007, Cambridge, MA, USA, pages 79–86, 2007. Cheng Huang, Huseyin Simitci, and et al. Erasure coding in windows azure storage. In Presented as part of the 2012 {USENIX} Annual Technical Conference ({USENIX}{ATC} 12), pages 15–26, 2012. Osama Khan, Randal C. Burns, James S. Plank, and Cheng Huang. In search of I/O-optimal recovery from disk failures. In 3rd USENIX Workshop on Hot Topics in Storage and File Systems, HotStorage’11, Portland, OR, USA, June 14, 2011, 2011. James L. Massey, Daniel J Costello, and Jørn Justesen. Polynomial weights and code
Maheswaran Sathiamoorthy, Megasthenis Asteris, and et al. Xoring elephants: Novel erasure codes for big data. arXiv preprint arXiv:1301.3791, 2013.
Zhao, Shum, Yang (CUHK(SZ) & USTC) Optimal LRCs ISIT 2020 16 / 16