Abstract
The concept of a locally repairable code (LRC) was introduced to protect the data from disk failures in large-scale storage systems. In this paper, we consider the LRCs with multiple disjoint repair sets and each repair set contains exactly one check symbol. By using several structures from combinatorial design theory, such as balanced incomplete block design, cyclic packing, group divisible design, near-Skolem sequence and Langford sequence, we construct several infinite classes of LRCs with the size of each repair set at most 3, which are optimal with respect to the bound proposed by Rawat et al. in 2016.
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Acknowledgements
This work is in part supported by NSFC (Nos.11601096, 11671103, 61672176), 2016GXNSF(Nos. FA380011, CA380021), 2018GXNSFAA138152, Guangxi Higher Institutions Program of Introducing 100 High-Level Overseas Talents, Research Fund of Guangxi Key Lab of Multi-source Information Mining & Security (16-B-01,18-B-01), Guangxi Collaborative Innovation Center of Multi-source Information Integration and Intelligent Processing, and the Guangxi “Bagui Scholar” Teams for Innovation and Research Project.
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Appendices
Appendix A: The proof of Lemma 4
In order to construct desired cyclic packings, we need the following definitions of a near-Skolem sequence and a Langford sequence.
Definition 5
Suppose that \(\mathcal {P}=\{(a_{i}, b_{i}) \ | \ 1 \leq i \leq u\}\) is a set of ordered pairs satisfying \(\bigcup \limits _{i=1}^{u}\{a_{i}, b_{i}\}=\{1,2,\cdots ,2u\}\).
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\(\mathcal {P}\)is a near-Skolem sequence of order u + 1 and defect m, m < u, denoted by NS(u + 1, m), if \(\bigcup \limits _{i=1}^{u}\{b_{i}-a_{i}\}=\{1,\cdots , m-1, m+1, \cdots , u+1\}\).
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\(\mathcal {P}\) is a Langford sequence of order u and defect m, m ≤ u, denoted by LS(u, m), if \(\bigcup \limits _{i=1}^{u}\{b_{i}-a_{i}\}=\{m, m+1,\cdots ,u+m-1\}\).
Example 3
The following sets \(\mathcal {P}_{N}\) and \(\mathcal {P}_{L}\) are NS(7,4) and LS(5,3), respectively.
Lemma 18
[11]
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An NS(u, m) exists if and only if
(i) u ≥ 2m − 1;
(ii) u ≡ 0,1 (mod 4) when m is odd, and u ≡ 2,3 (mod 4) when m is even.
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An LS(u, m) exists if and only if
(i) u ≥ 2m − 1;
(ii) u ≡ 0,1 (mod 4) when m is odd, u ≡ 0,3 (mod 4) when m is even.
Now, we will construct (k,3,1)-CPs by means of near-Skolem sequences and Langford sequences.
Lemma 19
Suppose that k ≡ 1,2,4,5,7,8,10,11 (mod 24) and k ≥ 49. Then there exists a (k,3,1)-CP \({\mathcal B}\) with \(|{\mathcal B}| = \lfloor \frac {k-1}{6}\rfloor -1\) and \(\{\pm 1,\pm 2\} \subseteq \)DL(\({\mathcal B}\)).
Proof
Let \(u=\lfloor \frac {k-1}{6}\rfloor \). Then u − 1 ≡ 0,1 (mod 4) and u − 1 ≥ 7. According to Lemma 18, there exists an LS(u − 1,4) {(ai, bi) | 1 ≤ i ≤ u − 1}. Let
Since
we have \({\varDelta }{\mathcal B} = Z_{k}\setminus \text {DL}^{(j)}({\mathcal B})\) where j ≡ k (mod 6), and
So \({\mathcal B}\) is a (k,3,1)-CP with \(\{\pm 1,\pm 2\} \subseteq \)DL(\({\mathcal B}\)). □
Lemma 20
Suppose that k ≡ 13,14,16,17,19,20,22,23 (mod 24) and k ≥ 259. Then there exists a (k,3,1)-CP \({\mathcal B}\)with \(|{\mathcal B}| = \lfloor \frac {k-1}{6}\rfloor -1\)and \(\{\pm 1,\pm 2\} \subseteq \)DL(\({\mathcal B}\)).
Proof
According to Lemma 18, there exists an LS(5,3) \(\{(a^{\prime }_{i}, b^{\prime }_{i}) \ | \ 1 \leq i \leq 5\}\). Then
Let \({\mathcal B}_{1}=\{\{0,a^{\prime }_{i}+7,b^{\prime }_{i}+7\} \ | \ 1\leq i\leq 5\}\).
Assume \(u=\lfloor \frac {k-1}{6}\rfloor \). Then u − 6 ≡ 0,1 (mod 4) and u − 6 ≥ 37. From Lemma 18, there exists an LS(u − 6,19) {(ai, bi) | 1 ≤ i ≤ u − 6}. Let
Since
we have \({\varDelta }{\mathcal B} = Z_{k}\setminus \text {DL}^{(j)}({\mathcal B})\) where j ≡ k (mod 6), and
So \({\mathcal B}\) is a (k,3,1)-CP with \(\{\pm 1,\pm 2\} \subseteq \)DL(\({\mathcal B}\)). □
For the rest values of k not being contained in Lemmas 19 and 20, we can search a corresponding cyclic packings by using a computer.
Lemma 21
Suppose that k = 6u + j, where u ∈{2, 3, 4, 5, 6, 7, 10, 11, 14, 15, 18, 19, 22, 23, 26, 27, 30, 31, 34, 35, 38, 39, 42} and j ∈{1,2,4,5}. There exists a (k,3,1)-CP \({\mathcal B}\)with \(|{\mathcal B}| = \lfloor \frac {k-1}{6}\rfloor -1\)and \(\{\pm 1,\pm 2\} \subseteq \)DL(\({\mathcal B}\)).
Proof
For u = 3 and j ∈{1,2,4,5}, the following \({\mathcal B}^{(u,j)}\) is the desired cyclic packing:
For any u≠ 3 and any j ∈{1,2,4,5}, a (k = 6u + j,3,1)-CP \({\mathcal B}^{(u)}\) is listed as following:
This completes the proof. □
The proof of Lemma 4 From Lemmas 19-21, we can obtain a (k,3,1)-CP \({\mathcal B} \) with \(|{\mathcal B}| =\lfloor \frac {k-1}{6}\rfloor -1\) and {± 1,± 2} \(\subseteq \) DL(\({\mathcal B}\)). Furthermore, since ± 4∉ DL\(({\mathcal B})\), there must exist a base block \(B \in {\mathcal B}\), such that ± 4 ∈ΔB. Then \({\mathcal B}^{\prime }\) = \({\mathcal B}\) ∖ B is a(k,3,1)-CP \({\mathcal B}^{\prime }\) with \(|{\mathcal B}^{\prime }| =\lfloor \frac {k-1}{6}\rfloor -2\) and \(\{\pm 1,\pm 2, \pm 4\} \subseteq \)DL(\({\mathcal B}^{\prime }\)).
Appendix B: The proof of Lemma 10
We first obtain the following lemma.
Lemma 22
Suppose that k ≡ 4 (mod 6). If there exists a (k − 1,3,1)-CP \({\mathcal B}\) with \(|{\mathcal B}|=\frac {k-4}{6}\) and {±y} \(\subseteq \) DL\(({\mathcal B})\), where y ∈{1,2}, then there exists a \((\frac {k-2}{2})\)-regular (k,{3,1∗},1)-packing.
Proof
Let \(u=\frac {k-4}{6}\), \(X= Z_{k-1}\bigcup \{\infty \}\), and \({\mathcal A}_{1} = \{ \{B+x\} \ | \ B \in {\mathcal B}, x \in Z_{k-1}\}\). Further, construct the following blocks:
Then \({\mathcal A} = {\mathcal A}_{1} \bigcup {\mathcal A}_{2}\) is the desired design. □
Based on the above lemma, in order to obtain the desired packings in this section, we need a (k − 1,3,1)-CP \({\mathcal B}\) with {±y} \(\subseteq \) DL\(({\mathcal B})\), where y ∈{1,2}. We now construct such cyclic packings in the next two lemmas.
Lemma 23
Suppose that k ≡ 4,22 (mod 24) and k ≥ 22. Then there exists a (k − 1,3,1)-CP \({\mathcal B}\)with \(|{\mathcal B}|=\frac {k-4}{6}\)and \(\{\pm 1\} \subseteq \)DL\(({\mathcal B})\).
Proof
Let \(u =\frac {k-4}{6}\). Then u ≡ 0,3 (mod 4) and u ≥ 3. According to Lemma 18, there exists an LS(u,2) {(ai, bi) | 1 ≤ i ≤ u}. Let \(B_{i}=\{0,a_{i}+u+1,b_{i}+u+1\}, 1\leq i\leq u, \ \text {and} \ {\mathcal B}= \bigcup \limits _{i=1}^{u}B_{i}\). Since
we have \({\varDelta }{\mathcal B} = Z_{6u+3} \setminus \{0, \pm 1\} = Z_{k-1} \setminus \{0, \pm 1\}\), which implies \({\mathcal B}\) is a (k − 1,3,1)-CP with \(\{ \pm 1\} \subseteq \) DL\(({\mathcal B})\). □
Lemma 24
Suppose that k ≡ 10,16 (mod 24) and k ≥ 16. Then there exists a (k − 1,3,1)-CP \({\mathcal B}\)with \(|{\mathcal B}|=\frac {k-4}{6}\)and \(\{ \pm 2\} \subseteq \)DL\(({\mathcal B})\).
Proof
Let \(u =\frac {k-4}{6}\). Then u + 1 ≡ 2,3 (mod 4) and u + 1 ≥ 3. From Lemma 18, there exists an NS(u + 1,2) {(ai, bi) | 1 ≤ i ≤ u}. Let Bi = {0, ai + u + 1, bi + u + 1},1 ≤ i ≤ u, and \({\mathcal B}= \bigcup \limits _{i=1}^{u}B_{i}\). Since
we have \({\varDelta }{\mathcal B} = Z_{6u+3} \setminus \{0, \pm 2\} = Z_{k-1} \setminus \{0, \pm 2\}\), which implies \({\mathcal B}\) is a (k − 1,3,1)-CP with \(\{ \pm 2\} \subseteq \) DL\(({\mathcal B})\). □
The proof of Lemma 10 If k > 10, we have a \(\frac {k-2}{2}\)-regular (k,{3,1∗},1)-packing from Lemmas 22-24. If k ∈{4,10}, we have a 1-regular (4,{3,1∗},1)-packing \({\mathcal A}^{(4,1)}\) and a 4-regular (10,{3,1∗},1)-packing \({\mathcal A}^{(10,4)}\) as following:
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Jiang, J., Cheng, M. Regular (k, R, 1)-packings with \(\max \limits {(R)}=3\) and their locally repairable codes. Cryptogr. Commun. 12, 1071–1089 (2020). https://doi.org/10.1007/s12095-020-00424-4
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DOI: https://doi.org/10.1007/s12095-020-00424-4