arXiv subjects

Sergey Yekhanin

Publications and source records attributed to Sergey Yekhanin.

27 records · Page 2Linked to original sources

arXiv2016

Maximally Recoverable Codes for Grid-like Topologies

The explosion in the volumes of data being stored online has resulted in distributed storage systems transitioning to erasure coding based schemes. Yet, the codes being deployed in practice are fairly short. In this work, we address what we view as the main coding theoretic barrier to deploying longer codes in storage: at large lengths, failures are not independent and correlated failures are inevitable. This motivates designing codes that allow quick data recovery even after large correlated failures, and which have efficient encoding and decoding. We propose that code design for distributed storage be viewed as a two-step process. The first step is choose a topology of the code, which incorporates knowledge about the correlated failures that need to be handled, and ensures local recovery from such failures. In the second step one specifies a code with the chosen topology by choosing coefficients from a finite field. In this step, one tries to balance reliability (which is better over larger fields) with encoding and decoding efficiency (which is better over smaller fields). This work initiates an in-depth study of this reliability/efficiency tradeoff. We consider the field-size needed for achieving maximal recoverability: the strongest reliability possible with a given topology. We propose a family of topologies called grid-like topologies which unify a number of topologies considered both in theory and practice, and prove a collection of results about maximally recoverable codes in such topologies including the first super-polynomial lower bound on the field size.

cs.IT↗

arXiv2016

New Constructions of SD and MR Codes over Small Finite Fields

Data storage applications require erasure-correcting codes with prescribed sets of dependencies between data symbols and redundant symbols. The most common arrangement is to have $k$ data symbols and $h$ redundant symbols (that each depends on all data symbols) be partitioned into a number of disjoint groups, where for each group one allocates an additional (local) redundant symbol storing the parity of all symbols in the group. A code as above is maximally recoverable, if it corrects all erasure patterns that are information theoretically correctable given the dependency constraints. A slightly weaker guarantee is provided by SD codes. One key consideration in the design of MR and SD codes is the size of the finite field underlying the code as using small finite fields facilitates encoding and decoding operations. In this paper we present new explicit constructions of SD and MR codes over small finite fields.

cs.IT↗

arXiv2013

Explicit Maximally Recoverable Codes with Locality

Consider a systematic linear code where some (local) parity symbols depend on few prescribed symbols, while other (heavy) parity symbols may depend on all data symbols. Local parities allow to quickly recover any single symbol when it is erased, while heavy parities provide tolerance to a large number of simultaneous erasures. A code as above is maximally-recoverable if it corrects all erasure patterns which are information theoretically recoverable given the code topology. In this paper we present explicit families of maximally-recoverable codes with locality. We also initiate the study of the trade-off between maximal recoverability and alphabet size.

cs.IT↗

arXiv2013

On the Locality of Codeword Symbols in Non-Linear Codes

Consider a possibly non-linear (n,K,d)_q code. Coordinate i has locality r if its value is determined by some r other coordinates. A recent line of work obtained an optimal trade-off between information locality of codes and their redundancy. Further, for linear codes meeting this trade-off, structure theorems were derived. In this work we give a new proof of the locality / redundancy trade-off and generalize structure theorems to non-linear codes.

cs.IT↗

arXiv2011

On the Locality of Codeword Symbols

Consider a linear [n,k,d]_q code C. We say that that i-th coordinate of C has locality r, if the value at this coordinate can be recovered from accessing some other r coordinates of C. Data storage applications require codes with small redundancy, low locality for information coordinates, large distance, and low locality for parity coordinates. In this paper we carry out an in-depth study of the relations between these parameters. We establish a tight bound for the redundancy n-k in terms of the message length, the distance, and the locality of information coordinates. We refer to codes attaining the bound as optimal. We prove some structure theorems about optimal codes, which are particularly strong for small distances. This gives a fairly complete picture of the tradeoffs between codewords length, worst-case distance and locality of information symbols. We then consider the locality of parity check symbols and erasure correction beyond worst case distance for optimal codes. Using our structure theorem, we obtain a tight bound for the locality of parity symbols possible in such codes for a broad class of parameter settings. We prove that there is a tradeoff between having good locality for parity checks and the ability to correct erasures beyond the minimum distance.

cs.IT↗

arXiv2010

Sets with large additive energy and symmetric sets

We show that for any set A in a finite Abelian group G that has at least c |A|^3 solutions to a_1 + a_2 = a_3 + a_4, where a_i belong A there exist sets A' in A and L in G, |L| \ll c^{-1} log |A| such that A' is contained in Span of L and A' has approximately c |A|^3 solutions to a'_1 + a'_2 = a'_3 + a'_4, where a'_i belong A'. We also study so-called symmetric sets or, in other words, sets of large values of convolution.

math.CO↗

arXiv2007

Locally Decodable Codes From Nice Subsets of Finite Fields and Prime Factors of Mersenne Numbers

A k-query Locally Decodable Code (LDC) encodes an n-bit message x as an N-bit codeword C(x), such that one can probabilistically recover any bit x_i of the message by querying only k bits of the codeword C(x), even after some constant fraction of codeword bits has been corrupted. The major goal of LDC related research is to establish the optimal trade-off between length and query complexity of such codes. Recently [Y] introduced a novel technique for constructing locally decodable codes and vastly improved the upper bounds for code length. The technique is based on Mersenne primes. In this paper we extend the work of [Y] and argue that further progress via these methods is tied to progress on an old number theory question regarding the size of the largest prime factors of Mersenne numbers. Specifically, we show that every Mersenne number m=2^t-1 that has a prime factor p>m^γyields a family of k(γ)-query locally decodable codes of length Exp(n^{1/t}). Conversely, if for some fixed k and all ε> 0 one can use the technique of [Y] to obtain a family of k-query LDCs of length Exp(n^ε); then infinitely many Mersenne numbers have prime factors arger than known currently.

cs.CC↗

arXiv2004

Improved Upper Bound for the Redundancy of Fix-Free Codes

A variable-length code is a fix-free code if no codeword is a prefix or a suffix of any other codeword. In a fix-free code any finite sequence of codewords can be decoded in both directions, which can improve the robustness to channel noise and speed up the decoding process. In this paper we prove a new sufficient condition of the existence of fix-free codes and improve the upper bound on the redundancy of optimal fix-free codes.

cs.IT↗

arXiv2004

Long Nonbinary Codes Exceeding the Gilbert - Varshamov Bound for any Fixed Distance

Let A(q,n,d) denote the maximum size of a q-ary code of length n and distance d. We study the minimum asymptotic redundancy ρ(q,n,d)=n-log_q A(q,n,d) as n grows while q and d are fixed. For any d and q<=d-1, long algebraic codes are designed that improve on the BCH codes and have the lowest asymptotic redundancy ρ(q,n,d) <= ((d-3)+1/(d-2)) log_q n known to date. Prior to this work, codes of fixed distance that asymptotically surpass BCH codes and the Gilbert-Varshamov bound were designed only for distances 4,5 and 6.

cs.IT↗