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 (CUHK(SZ) & USTC) Optimal LRCs ISIT 2020 1 / 16
The Large-Scale Distributed Storage System 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 of erasure coding. 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
The Large-Scale Distributed Storage System 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 of erasure coding. 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
The Large-Scale Distributed Storage System 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 of erasure coding. 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
The Large-Scale Distributed Storage System 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 of erasure coding. 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
The Locally Repairable Codes Given a block code C of length n over F q , a position i has locality r if there exists λ j t ∈ F q , where 1 ≤ t ≤ r and j t � = i , such that for each codeword ( c 0 , c 1 , · · · , c n − 1 ) ∈ C r � c i = λ j t c j t (1) t =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 Locally Repairable Codes Given a block code C of length n over F q , a position i has locality r if there exists λ j t ∈ F q , where 1 ≤ t ≤ r and j t � = i , such that for each codeword ( c 0 , c 1 , · · · , c n − 1 ) ∈ C r � c i = λ j t c j t (1) t =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 Locally Repairable Codes Given a block code C of length n over F q , a position i has locality r if there exists λ j t ∈ F q , where 1 ≤ t ≤ r and j t � = i , such that for each codeword ( c 0 , c 1 , · · · , c n − 1 ) ∈ C r � c i = λ j t c j t (1) t =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
Optimal LRCs The Singleton-like bound for an [ n, k, d ] code with locality r is � k � n − k ≥ + d − 2 . (2) r 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
Constacyclic codes For a unit λ ∈ F q , the λ -constacyclic shift τ λ on F n q is τ λ ( c 0 , c 1 , . . . , c n − 1 ) = ( λc n − 1 , c 0 , . . . , c n − 2 ) . (3) A linear code C is said to be λ -constacyclic if τ λ ( C ) = C . C is a cyclic code if λ = 1 . Polynomial representation: c = ( c 0 , c 1 , . . . , c n − 1 ) ↔ c ( x ) = c 0 + c 1 x + · · · + c n − 1 x n − 1 (4) τ λ ( c ) ↔ xc ( x ) ∈ F q [ x ] / � x n − λ � (5) A λ -constacyclic code ↔ An ideal of F q [ x ] / � x n − λ � (6) Zhao, Shum, Yang (CUHK(SZ) & USTC) Optimal LRCs ISIT 2020 5 / 16
Constacyclic codes For a unit λ ∈ F q , the λ -constacyclic shift τ λ on F n q is τ λ ( c 0 , c 1 , . . . , c n − 1 ) = ( λc n − 1 , c 0 , . . . , c n − 2 ) . (3) A linear code C is said to be λ -constacyclic if τ λ ( C ) = C . C is a cyclic code if λ = 1 . Polynomial representation: c = ( c 0 , c 1 , . . . , c n − 1 ) ↔ c ( x ) = c 0 + c 1 x + · · · + c n − 1 x n − 1 (4) τ λ ( c ) ↔ xc ( x ) ∈ F q [ x ] / � x n − λ � (5) A λ -constacyclic code ↔ An ideal of F q [ x ] / � x n − λ � (6) Zhao, Shum, Yang (CUHK(SZ) & USTC) Optimal LRCs ISIT 2020 5 / 16
Constacyclic codes For a λ -constacyclic code C , there exists a unique polynomial � ( x − α i ) g ( x ) = (7) i ∈ T 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 = { α i | i ∈ T } is called the zeros of C . Zhao, Shum, Yang (CUHK(SZ) & USTC) Optimal LRCs ISIT 2020 6 / 16
Known Constructions of Optimal Cylic LRCs Most existing constructions of optimal cyclic LRCs use zero structures: 1 If Z ⊃ { α i ( r +1) | 0 ≤ i ≤ n r +1 − 1 } , then C has locality r . 2 If Z ⊃ { α b , α b +1 , · · · , α b + 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
Limitations of the Existing Constructions of Optimal Cylic LRCs 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
Limitations of the Existing Constructions of Optimal Cylic LRCs 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
Our Results We mainly consider the λ -constacyclic codes of length p s over finite field F p m , which are the ideals of F p m [ x ] / � x p s − 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 (2 s , 2 s − 1 − 1 , 4 , 1) 2 m , where s ≥ 2 ; C.2 ( p s , p s − p s − k − 1 , 2 , p k +1 − 1) p m , where 1 ≤ k ≤ s − 1 and s ≥ 2 ; C.3 ( p s , p s − 2 , 2 , p s − p s − 1 − 1) p m , where p ≥ 3 and s ≥ 2 ; C.4 ( p s , p s − p s − 1 − 1 , 3 , p − 1) p m , where p ≥ 3 and s ≥ 2 ; C.5 ( p s , 1 , p s , 1) p m , where s ≥ 2 ; C.6 ( p s , p s − p s − 1 , 2 , p − 1) p m ; C.7 ( p, p − t − 1 , t + 2 , p − t − 1) p m , where 1 ≤ t ≤ p − 2 and p ≥ 3 . Zhao, Shum, Yang (CUHK(SZ) & USTC) Optimal LRCs ISIT 2020 9 / 16
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