Combinatorial and LP bounds for LRC codes

Hu, Sihuang; Tamo, Itzhak; Barg, Alexander

doi:10.1109/isit.2016.7541451

Cited by 9 publications

(15 citation statements)

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“…where the 'Min' is taken over (10). Note that 2n−l(r+2) ≥ 0, so a necessary condition for optimizing (8) is that l j=1 r j = 2n − l(r + 2). Then the optimization can be restricted to the condition (9), and thus the theorem holds.…”

Section: (9)mentioning

confidence: 99%

“…Overall, most of the bounds derived so far for LRCs over particular finite fields either depend on undetermined parameters in coding theory, e.g., the C-M bound, or rely on solving optimization problems under concrete code parameters, e.g., the LP bound in [8]. And the constructions of binary LRCs with good parameters mostly restrict to specific values of d and r. Much work remains undone for LRCs over particular finite fields.…”

Section: Introductionmentioning

confidence: 99%

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Bounds and constructions for linear locally repairable codes over binary fields

Wang

Zhang

Lin

2017

2017 IEEE International Symposium on Information Theory (ISIT)

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For binary [n, k, d] linear locally repairable codes (LRCs), two new upper bounds on k are derived. The first one applies to LRCs with disjoint local repair groups, for general values of n, d and locality r, containing some previously known bounds as special cases. The second one is based on solving an optimization problem and applies to LRCs with arbitrary structure of local repair groups. Particularly, an explicit bound is derived from the second bound when d ≥ 5. A specific comparison shows this explicit bound outperforms the Cadambe-Mazumdar bound for 5 ≤ d ≤ 8 and large values of n. Moreover, a construction of binary linear LRCs with d ≥ 6 attaining our second bound is provided.

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Section: (9)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Bounds and constructions for linear locally repairable codes over binary fields

Wang

Zhang

Lin

2017

2017 IEEE International Symposium on Information Theory (ISIT)

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“…. , α n−1 }, where α is a primitive n-th root of unity, we construct a linear k-dimensional code C using the evaluation map (6). Using this representation as the starting point, we observe that C is a cyclic code of length n. Generally, a cyclic code is an ideal in the ring F q [x]/(x n −1) which is generated by a polynomial g(x) such that g(x)|(x n −1).…”

Section: Cyclic Q-ary Lrc Codesmentioning

confidence: 99%

Cyclic LRC codes, binary LRC codes, and upper bounds on the distance of cyclic codes

Tamo

Barg

Goparaju

et al. 2016

IJICOT

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Abstract:We consider linear cyclic codes with the locality property or locally recoverable codes (LRC codes). A family of LRC codes that generalises the classical construction of Reed-Solomon codes was constructed in a recent paper by Tamo and Barg (IEEE Transactions on Information Theory, No. 8, 2014). In this paper, we focus on distance-optimal cyclic codes that arise from this construction. We give a characterisation of these codes in terms of their zeros and observe that there are many equivalent ways of constructing optimal cyclic LRC codes over a given field. We also study subfield subcodes of cyclic LRC codes (BCH-like LRC codes) and establish several results about their locality and minimum distance. The locality parameter of a cyclic code is related to the Copyright c 2016 Inderscience Enterprises Ltd. 346 I. Tamo et al.dual distance of this code, and we phrase our results in terms of upper bounds on the dual distance.Keywords: cyclic LRC codes; irreducible cyclic codes; subfield subcodes; zeros of the code.Reference to this paper should be made as follows: Tamo, I., Barg, A., Goparaju, S. and Calderbank, R. (2016) Mathematics at Princeton University, and before that he was Vice President for Research at AT&T. Innovations by him are incorporated in a progression of voiceband modem standards that moved communications practice close to the Shannon limit. Together with Peter Shor and colleagues at AT&T Labs, he showed that good quantum error correcting codes exist and developed the group theoretic framework for quantum error correction. He is a co-inventor of space-time codes for wireless communication, where correlation of signals across different transmit antennas leads to reliable transmission. This paper is a revised and expanded version of a paper entitled 'Cyclic LRC codes and their subfield subcodes', presented at

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“…The bound becomes stronger if we consider disjoint repair groups. This paper is a result of merging and developing the papers by S. Hu, I. Tamo, and A. Barg [25] and by A. Agarwal and A. Mazumdar [26], both devoted to the problem of deriving alphabet-dependent bounds on LRC codes.…”

Section: Introductionmentioning

confidence: 99%

Combinatorial Alphabet-Dependent Bounds for Locally Recoverable Codes

Agarwal

Barg

et al. 2018

IEEE Trans. Inform. Theory

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Locally recoverable (LRC) codes have recently been a focus point of research in coding theory due to their theoretical appeal and applications in distributed storage systems. In an LRC code, any erased symbol of a codeword can be recovered by accessing only a small number of other symbols. For LRC codes over a small alphabet (such as binary), the optimal rate-distance trade-off is unknown. We present several new combinatorial bounds on LRC codes including the locality-aware sphere packing and Plotkin bounds. We also develop an approach to linear programming (LP) bounds on LRC codes. The resulting LP bound gives better estimates in examples than the other upper bounds known in the literature. Further, we provide the tightest known upper bound on the rate of linear LRC codes with a given relative distance, an improvement over the previous best known bounds.

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Combinatorial and LP bounds for LRC codes

Cited by 9 publications

References 10 publications

Bounds and constructions for linear locally repairable codes over binary fields

Bounds and constructions for linear locally repairable codes over binary fields

Cyclic LRC codes, binary LRC codes, and upper bounds on the distance of cyclic codes

Combinatorial Alphabet-Dependent Bounds for Locally Recoverable Codes

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