Paper Info
Title
Maximally Recoverable LRCs: A Field Size Lower Bound and Constructions for Few Heavy Parities
Abstract
The explosion in the volumes of data being stored online has resulted in distributed storage ‘s transitioning to erasure coding based schemes. Local Reconstruction Codes (LRCs) have emerged as the codes of choice for these applications. These codes can correct a small number of erasures (which is the typical case) by accessing only a small number of remaining coordinates. An <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(n,r,h,a,q)$ </tex-math></inline-formula> -LRC is a linear code over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {F}_{q}$ </tex-math></inline-formula> of length <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> , whose codeword symbols are partitioned into <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$g=n/r$ </tex-math></inline-formula> local groups each of size <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$r$ </tex-math></inline-formula> . Each local group has <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$a$ </tex-math></inline-formula> local parity checks that allow recovery of up to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$a$ </tex-math></inline-formula> erasures within the group by reading the unerased symbols in the group. There are a further <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$h$ </tex-math></inline-formula> “heavy” parity checks to provide fault tolerance from more global erasure patterns. Such an LRC is Maximally Recoverable (MR), if it corrects all erasure patterns which are information-theoretically correctable under the stipulated structure of local and global parity checks, namely patterns with up to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$a$ </tex-math></inline-formula> erasures in each local group and an additional <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$h$ </tex-math></inline-formula> (or fewer) erasures anywhere in the codeword. The existing constructions require fields of size <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n^{\Omega (h)}$ </tex-math></inline-formula> while no superlinear lower bounds were known for any setting of parameters. Is it possible to get linear field size similar to the related MDS codes (e.g., Reed-Solomon codes)? In this work, we answer this question by showing superlinear lower bounds on the field size of MR-LRCs. When <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$a,h$ </tex-math></inline-formula> are constant and the number of local groups <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$g \geqslant h$ </tex-math></inline-formula> , while <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$r$ </tex-math></inline-formula> may grow with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> , our lower bound simplifies to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q \geqslant \Omega _{a,h}\left ({n\cdot r^{\min \{a,h-2\}}}\right)$ </tex-math></inline-formula> . MR-LRCs deployed in practice have a small number of global parities, typically <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$h=2,3$ </tex-math></inline-formula> . We complement our lower bounds by giving constructions with small field size for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$h \leqslant 3$ </tex-math></inline-formula> . When <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$h=2$ </tex-math></inline-formula> , we give a linear field size construction, whereas previous constructions required quadratic field size in some parameter ranges. Note that our lower bound is superlinear only if <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$h \geqslant 3$ </tex-math></inline-formula> . When <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$h=3$ </tex-math></inline-formula> , we give a construction with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$O(n^{3})$ </tex-math></inline-formula> field size, whereas previous constructions needed <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n^{\Theta (a)}$ </tex-math></inline-formula> field size. This makes the choices <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$r=3, a=1, h=3$ </tex-math></inline-formula> the next simplest non-trivial setting to investigate regarding the existence of MR-LRCs over fields of near-linear size. We answer this question in the positive via a novel approach based on elliptic curves and arithmetic progression free sets.
Year
DOI
Venue
2020
10.1109/TIT.2020.2990981
IEEE Transactions on Information Theory
Keywords
DocType
Volume
Parity check codes,Linear codes,Reed-Solomon codes,Explosions,Decoding,Distributed databases
Journal
66
Issue
ISSN
Citations 
10
0018-9448
5
PageRank 
References 
Authors
0.43
0
3
Name
Order
Citations
PageRank
Sivakanth Gopi1255.63
V. Guruswami23205247.96
S. Yekhanin3323.12