Constructions of Optimal Binary Locally Repairable Codes With Multiple Repair Groups

Hao, Jie; Xia, Shu-Tao

doi:10.1109/lcomm.2016.2539160

Cited by 33 publications

(28 citation statements)

References 15 publications

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“…The minimum Hamming distance of array LDPC codes is known for certain sets of parameters (see, e.g., [44] and references therein). Codes C 6 and C 7 in Table II are optimal LRCs based on array LDPC codes constructed using [43,Constr. 3] and having information locality 11.…”

Section: Numerical Resultsmentioning

confidence: 99%

Achieving Maximum Distance Separable Private Information Retrieval Capacity With Linear Codes

Kumar

Lin

Rosnes

et al. 2019

IEEE Trans. Inform. Theory

101

100

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We propose three private information retrieval (PIR) protocols for distributed storage systems (DSSs) where data is stored using an arbitrary linear code. The first two protocols, named Protocol 1 and Protocol 2, achieve privacy for the scenario with noncolluding nodes. Protocol 1 requires a file size that is exponential in the number of files in the system, while Protocol 2 requires a file size that is independent of the number of files and is hence simpler. We prove that, for certain linear codes, Protocol 1 achieves the maximum distance separable (MDS) PIR capacity, i.e., the maximum PIR rate (the ratio of the amount of retrieved stored data per unit of downloaded data) for a DSS that uses an MDS code to store any given (finite and infinite) number of files, and Protocol 2 achieves the asymptotic MDS-PIR capacity (with infinitely large number of files in the DSS). In particular, we provide a necessary and a sufficient condition for a code to achieve the MDS-PIR capacity with Protocols 1 and 2 and prove that cyclic codes, Reed-Muller (RM) codes, and a class of distance-optimal local reconstruction codes achieve both the finite MDS-PIR capacity (i.e., with any given number of files) and the asymptotic MDS-PIR capacity with Protocols 1 and 2, respectively. Furthermore, we present a third protocol, Protocol 3, for the scenario with multiple colluding nodes, which can be seen as an improvement of a protocol recently introduced by Freij-Hollanti et al.. Similar to the noncolluding case, we provide a necessary and a sufficient condition to achieve the maximum possible PIR rate of Protocol 3. Moreover, we provide a particular class of codes that is suitable for this protocol and show that RM codes achieve the maximum possible PIR rate for the protocol. For all three protocols, we present an algorithm to optimize their PIR rates.

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Section: Numerical Resultsmentioning

confidence: 99%

Achieving Maximum Distance Separable Private Information Retrieval Capacity With Linear Codes

Kumar

Lin

Rosnes

et al. 2019

IEEE Trans. Inform. Theory

101

100

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“…In [12], a construction of optimal (in terms of minimum distance) binary LRCs with multiple repair groups was given. In particular, in [12,Constr. 3], a construction based on array low-density parity-check (LDPC) codes was provided.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…The minimum distance of array LDPC codes is known for certain sets of parameters (see, e.g., [13], and references therein). Codes C 6 and C 7 in Table I are optimal LRCs based on array LDPC codes constructed using [12,Constr. 3] and having a locality of 11.…”

Section: Numerical Resultsmentioning

confidence: 99%

Private information retrieval in distributed storage systems using an arbitrary linear code

Kumar

Rosnes

Amat

2017

2017 IEEE International Symposium on Information Theory (ISIT)

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We propose an information-theoretic private information retrieval (PIR) scheme for distributed storage systems where data is stored using a linear systematic code of rate R > 1/2. The proposed scheme generalizes the PIR scheme for data stored using maximum distance separable codes recently proposed by Tajeddine and El Rouayheb for the scenario of a single spy node. We further propose an algorithm to optimize the communication price of privacy (cPoP) using the structure of the underlying linear code. As an example, we apply the proposed algorithm to several distributed storage codes, showing that the cPoP can be significantly reduced by exploiting the structure of the distributed storage code.

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“…Since k * = 5 and ⌈k/k * ⌉ = m − 1. Taking j = m − 2 in bound (2), d d (4) opt [30,3] 22 by the Griesmer bound, so d = 22 and the bound is met. ✷ Let us compare next EII codes with s t > 0 with II codes (i.e., s t = 0).…”

Section: Eii Codes Achieving Bound (2) and Comparison With Other mentioning

confidence: 99%

Extended Integrated Interleaved Codes Over Any Field With Applications to Locally Recoverable Codes

Blaum

2020

IEEE Trans. Inform. Theory

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Integrated Interleaved (II) and Extended Integrated Interleaved (EII) codes are a versatile alternative for Locally Recoverable (LRC) codes, since they require fields of relatively small size. II and EII codes are generally defined over Reed-Solomon type of codes. A new comprehensive definition of EII codes is presented, allowing for EII codes over any field, and in particular, over the binary field GF(2). The traditional definition of II and EII codes is shown to be a special case of the new definition. Improvements over previous constructions of LRC codes, in particular, for binary codes, are given, as well as cases meeting an upper bound on the minimum distance. Properties of the codes are presented as well, in particular, an iterative decoding algorithm on rows and columns generalizing the iterative decoding algorithm of product codes. Two applications are also discussed: one is finding a systematic encoding of EII codes such that the parity symbols have a balanced distribution on rows, and the other is the problem of ordering the symbols of an EII code such that the maximum length of a correctable burst is achieved.Constructions of LRC codes involve different issues and tradeoffs, like the size of the field and optimality criteria. The same is true for EPC codes, of which, as we have seen above, LRC codes are a special case. In particular, one goal is to keep the size of the required finite field small, since operations over a small field have less complexity than ones over a larger field due to the smaller look-up tables involved. For example, Integrated Interleaved (II) codes [32], [68] over GF(q), where q > max{m, n}, were proposed in [7] as LRC codes. The construction in [45], [46] (stair codes) reduces field size when failures are correlated. Extended Integrated Interleaved (EII) codes [8] unify product codes and II codes.As is the case with LRC codes, construction of EPC codes involves optimality issues. For example, LRC codes optimizing the minimum distance were presented in [67]. Except for special cases, II codes are not optimal as LRC codes, but the codes in [67] require a field of size at least mn. The same happens with EII codes: except for special cases, they do not optimize the minimum distance [8].There are stronger criteria for optimization than the minimum distance in LRC codes. For example, PMDS codes [5], [9], [19], [23], [33], [35] satisfy the Maximally Recoverable (MR) property [23], [25]. The definition of the MR property is extended for EPC codes in [25], but it turns out that EPC codes with the MR property are difficult to obtain. For example, in [25] it was proven that an EPC code EP(n, 1; n, 1; 1) (i.e., one vertical and one horizontal parity per column and row and one extra parity) with the MR property requires a field whose size is superlinear on n.The EII codes presented in [8], of which II codes [7], [68], [73], [76], [77] are special cases, in general assume a field GF(q) such that q > max{m, n} and the individual codes are RS type of codes. An exception occurs in [73], where b...

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Constructions of Optimal Binary Locally Repairable Codes With Multiple Repair Groups

Cited by 33 publications

References 15 publications

Achieving Maximum Distance Separable Private Information Retrieval Capacity With Linear Codes

Achieving Maximum Distance Separable Private Information Retrieval Capacity With Linear Codes

Private information retrieval in distributed storage systems using an arbitrary linear code

Extended Integrated Interleaved Codes Over Any Field With Applications to Locally Recoverable Codes

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