A Construction of Maximally Recoverable Codes with Order-Optimal Field Size

Cai, Han; Miao, Ying; Schwartz, Moshe; Tang, Xiaohu

doi:10.48550/arxiv.2011.13606

Cited by 6 publications

(5 citation statements)

References 54 publications

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“…Related Work. Shortly before we published our results, we learned that [CMST20] have independently obtained a result analogous to Theorem 1.3 with a very similar construction. They construct (n, r, h, a, q)-MR LRCs with a field size of…”

Section: Our Resultsmentioning

confidence: 96%

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Improved Maximally Recoverable LRCs using Skew Polynomials

Gopi¹,

Guruswami²

2020

Preprint

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An (n, r, h, a, q)-LRC is a linear code over F q of length n, whose codeword symbols are partitioned into n/r local groups each of size r. Each local group satisfies 'a' local parity checks to recover from 'a' erasures in that local group and there are further h global parity checks to provide fault tolerance from more global erasure patterns. Such an LRC is Maximally Recoverable (MR), if it can correct all erasure patterns which are information-theoretically correctable given this structure-these are precisely patterns with up to 'a' erasures in each local group and an additional h erasures anywhere in the codeword.We give an explicit construction of (n, r, h, a, q)-MR LRCs with field size q bounded by (O (max{r, n/r})) min{h,r−a} . This significantly improves upon known constructions in most parameter ranges. Moreover, it matches the best known lower bound from [GGY20] in an interesting special case when r = Θ( √ n) and h, a are constants with h a + 2, achieving the optimal field size of Θ a,h (n h/2 ). Our construction is based on the theory of skew polynomials.

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Section: Our Resultsmentioning

confidence: 96%

“…Soon after [CMST20], two more constructions of MR LRCs were published by [Mar20] with the following field sizes:…”

Section: Our Resultsmentioning

confidence: 99%

Improved Maximally Recoverable LRCs using Skew Polynomials

Gopi¹,

Guruswami²

2020

Preprint

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“…In [19], lower bounds on the field size requirement for MR codes were introduced. For explicit constructions of MR codes, the reader may refer to [3], [8], [15], [16], [18], [20], [28]. Notably, there are MR codes have order-optimal field size (with respect to the bound in [19]): [3] for a single global parity check (h = 1), [4], [19] for h = 2, [16] for h = 3 and δ = 2, and [8], [18] for h δ + 1 a constant, and n = Θ(r 2 ).…”

Section: Introductionmentioning

confidence: 99%

A Bound on the Minimal Field Size of LRCs, and Cyclic MR Codes That Attain It

Han¹,

Schwartz²

2022

Preprint

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We prove a new lower bound on the field size of locally repairable codes (LRCs). Additionally, we construct maximally recoverable (MR) codes which are cyclic. While a known construction for MR codes has the same parameters, it produces noncyclic codes. Furthermore, we prove necessary and sufficient conditions that specify when the known non-cyclic MR codes may be permuted to become cyclic, thus proving our construction produces cyclic MR codes with new parameters. Furthermore, using our new bound on the field size, we show that the new cyclic MR codes have optimal field size in certain cases. Other known LRCs are also shown to have optimal field size in certain cases.

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“…, as well as a number of constructions in [7] with comparable parameters. We refer the reader to the introduction of [6,7], or [3], Table 1, for a more detailed overview which also covers the entire range of possible parameters a, r, and h. As remarked in [7], most of the known constructions require alphabets of size q that depend exponentially on h. One exception where the minimum alphabet size is independent of h is the construction of [8] which requires q = O(max(n/(r + 1), r + 1)) r . This is larger than the result in Theorem 1 above both for fixed and growing r. In summary, the code family constructed in this paper improves upon the known results in terms of the required field size.…”

Section: Introductionmentioning

confidence: 99%