Almost affine locally repairable codes and matroid theory

Westerbäck, Thomas; Ernvall, Toni; Hollanti, Camilla

doi:10.1109/itw.2014.6970906

Cited by 10 publications

(6 citation statements)

References 18 publications

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“…This approach may be useful for functional repair code requiring repair-by-transfer ( [20,1,22]), where the nodes contributing information for repair do not perform any computations. There has also been studies of locally repairable codes via matroid theory ( [27,28]) which may also be of interest for functional repair codes.…”

Section: Further Workmentioning

confidence: 99%

Functional repair codes: a view from projective geometry

Paterson

2019

Des. Codes Cryptogr.

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Storage codes are used to ensure reliable storage of data in distributed systems. Here we consider functional repair codes, where individual storage nodes that fail may be repaired efficiently and the ability to recover original data and to further repair failed nodes is preserved. There are two predominant approaches to repair codes: a coding theoretic approach and a vector space approach. We explore the relationship between the two and frame the later in terms of projective geometry. We find that many of the constructions proposed in the literature can be seen to arise from natural and well-studied geometric objects, and that this perspective gives a framework that provides opportunities for generalisations and new constructions that can lead to greater flexibility in trade-offs between various desirable properties. We also frame the cut-set bound obtained from network coding in terms of projective geometry.We explore the notion of strictly functional repair codes, for which there exist nodes that cannot be replaced exactly. Currently only one known example is given in the literature, due to Hollmann and Poh. We examine this phenomenon from a projective geometry point of view, and discuss how strict functionality can arise.Finally, we consider the issue that the view of a repair code as a collection of sets of vector/projective subspaces is recursive in nature and makes it hard to visualise what a collection of nodes looks like and how one might approach a construction. Here we provide another view of using directed graphs that gives us non-recursive criteria for determining whether a family of collections of subspaces constitutes a function, exact, or strictly functional repair code, which may be of use in searching for new codes with desirable properties.

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Section: Further Workmentioning

confidence: 99%

Functional repair codes: a view from projective geometry

Paterson

2019

Des. Codes Cryptogr.

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“…We summarize previous constructions of optimal LRC codes in [23,29,13,15,25,26,27,2,20,19,4,16] as follows.…”

Section: Known Lrc Constructionsmentioning

confidence: 99%

“…1. Binary LRC codes over F 2 with large lengths: In [29] an almost optimal binary LRC code with n = 15, k = 10, r = 6 and d = 4 < 15 − 10 + 2 − ⌈ 10 6 ⌉ = 5 was constructed. In [13] a family of optimal binary cyclic LRC codes satisfying n = 2 m − 1 for some positive integer m, r + 1|n, d = 2 was constructed.…”

Section: N > Qmentioning

confidence: 99%

Long Optimal or Small-Defect LRC Codes with Unbounded Minimum Distances

Chen¹,

Weng²,

Luo³

2019

Preprint

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For a linear locally recoverable code (LRC) with length n, dimension k and locality r its minimum distance d satisfies d ≤ n−k+2−⌈ k r ⌉. A code attaining this bound is called optimal. Many families of optimal locally recoverable codes have been constructed by using different techniques in finite fields or algebraic curves. However in previous constructions of length n >> q optimal LRC codes minimum distances are only few constants smaller than 9. No optimal LRC code over a general finite field F q with the length n ∼ q 2 and the minimum distance d ≥ 9 has been constructed. In this paper we present a general construction of optimal LRC codes over arbitrary finite fields. Over any given finite field F q , for any given r ∈ {1, 2, . . . , q − 1} and given d satisfying 3 ≤ d ≤ min{r + 1, q + 1 − r}, we construct explicitly an optimal LRC code with length n = q(r + 1), locality r and minimum distance d. We also gives an asymptotic bound for q-ary r ≤ q − 1-locality LRC codes better than the previously konwn bound. Many long r-locality LRC codes with small defect s = n−k+2−⌈ k r ⌉−d are also constructed.

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“…See e.g. [17] An important subclass of almost affine codes are linear codes over finite fields F q . A bigger class consists of affine codes, which are translates of linear codes within their ambient space.…”

Section: Introductionmentioning

confidence: 99%