
Constructions of maximally recoverable local reconstruction codes via function fields
Local Reconstruction Codes (LRCs) allow for recovery from a small number...
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On Maximally Recoverable Local Reconstruction Codes
In recent years the explosion in the volumes of data being stored online...
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Maximally recoverable local reconstruction codes from subspace direct sum systems
Maximally recoverable local reconstruction codes (MR LRCs for short) hav...
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Universal and Dynamic Locally Repairable Codes with Maximal Recoverability via SumRank Codes
Locally repairable codes (LRCs) are considered with equal or unequal loc...
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Correctable Erasure Patterns in Product Topologies
Locality enables storage systems to recover failed nodes from small subs...
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Lower Bounds for Maximally Recoverable Tensor Code and Higher Order MDS Codes
An (m,n,a,b)tensor code consists of mΓ n matrices whose columns satisfy...
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Constructions of Locally Recoverable Codes which are Optimal
We provide a Galois theoretical framework which allows to produce good p...
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Improved Maximally Recoverable LRCs using Skew Polynomials
An (n,r,h,a,q)Local Reconstruction Code is a linear code over π½_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 offers the best blend of locality and global erasure resilience β namely it can correct all erasure patterns whose recovery is informationtheoretically feasible given the locality structure (these are precisely patterns with up to `a' erasures in each local group and an additional h erasures anywhere in the codeword). Random constructions can easily show the existence of MR LRCs over very large fields, but a major algebraic challenge is to construct MR LRCs, or even show their existence, over smaller fields, as well as understand inherent lower bounds on their field size. 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,ra}. This improves upon known constructions in many relevant parameter ranges. Moreover, it matches the lower bound from Gopi et al. (2020) in an interesting range of parameters where r=Ξ(β(n)), ra=Ξ(β(n)) and h is a fixed constant with hβ€ a+2, achieving the optimal field size of Ξ_h(n^h/2). Our construction is based on the theory of skew polynomials. We believe skew polynomials should have further applications in coding and complexity theory; as a small illustration we show how to capture algebraic results underlying list decoding folded ReedSolomon and multiplicity codes in a unified way within this theory.
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