Results

**1 - 4**of**4**### High rate locally-correctable and locally-testable codes with sub-polynomial query complexity∗

, 2015

"... In this work, we construct the first locally-correctable codes (LCCs), and locally-testable codes (LTCs) with constant rate, constant relative distance, and sub-polynomial query com-plexity. Specifically, we show that there exist binary LCCs and LTCs with block length n, constant rate (which can eve ..."

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In this work, we construct the first locally-correctable codes (LCCs), and locally-testable codes (LTCs) with constant rate, constant relative distance, and sub-polynomial query com-plexity. Specifically, we show that there exist binary LCCs and LTCs with block length n, constant rate (which can even be taken arbitrarily close to 1), constant relative distance, and query complexity exp(Õ( log n)). Previously such codes were known to exist only with Ω(nβ) query complexity (for constant β> 0), and there were several, quite different, constructions known. Our codes are based on a general distance-amplification method of Alon and Luby [AL96]. We show that this method interacts well with local correctors and testers, and obtain our main results by applying it to suitably constructed LCCs and LTCs in the non-standard regime of sub-constant relative distance. Along the way, we also construct LCCs and LTCs over large alphabets, with the same query complexity exp(Õ(

### High-rate locally-correctable and locally-testable codes with sub-polynomial query complexity∗

, 2015

"... In this work, we construct the first locally-correctable codes (LCCs), and locally-testable codes (LTCs) with constant rate, constant relative distance, and sub-polynomial query com-plexity. Specifically, we show that there exist binary LCCs and LTCs with block length n, constant rate (which can eve ..."

Abstract
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In this work, we construct the first locally-correctable codes (LCCs), and locally-testable codes (LTCs) with constant rate, constant relative distance, and sub-polynomial query com-plexity. Specifically, we show that there exist binary LCCs and LTCs with block length n, constant rate (which can even be taken arbitrarily close to 1), constant relative distance, and query complexity exp(Õ( log n)). Previously such codes were known to exist only with Ω(nβ) query complexity (for constant β> 0), and there were several, quite different, constructions known. Our codes are based on a general distance-amplification method of Alon and Luby [AL96]. We show that this method interacts well with local correctors and testers, and obtain our main results by applying it to suitably constructed LCCs and LTCs in the non-standard regime of sub-constant relative distance. Along the way, we also construct LCCs and LTCs over large alphabets, with the same query complexity exp(Õ(

### Batch Codes through Dense Graphs without Short Cycles

, 2014

"... Consider a large database of n data items that need to be stored using m servers. We study how to encode information so that a large number k of read requests can be performed in parallel while the rate remains constant (and ideally approaches one). This problem is equivalent to the design of multis ..."

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Consider a large database of n data items that need to be stored using m servers. We study how to encode information so that a large number k of read requests can be performed in parallel while the rate remains constant (and ideally approaches one). This problem is equivalent to the design of multiset Batch Codes introduced by Ishai, Kushilevitz, Ostrovsky and Sahai [17]. We give families of multiset batch codes with asymptotically optimal rates of the form 1−1/poly(k) and a number of servers m scaling polynomially in the number of read requests k. An advantage of our batch code constructions over most previously known multiset batch codes is explicit and deterministic decoding algorithms and asymptotically optimal fault tolerance. Our main technical innovation is a graph-theoretic method of designing multiset batch codes using dense bipartite graphs with no small cycles. We modify prior graph constructions of dense, high-girth graphs to obtain our batch code results. We achieve close to optimal tradeoffs between the parameters for bipartite graph based batch codes.

### High-rate codes with sublinear-time decoding

"... Locally decodable codes are error-correcting codes that admit efficient decoding algorithms; any bit of the original message can be recovered by looking at only a small number of locations of a corrupted codeword. The tradeoff between the rate of a code and the locality/efficiency of its decoding al ..."

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Locally decodable codes are error-correcting codes that admit efficient decoding algorithms; any bit of the original message can be recovered by looking at only a small number of locations of a corrupted codeword. The tradeoff between the rate of a code and the locality/efficiency of its decoding algorithms has been well studied, and it has widely been suspected that nontrivial locality must come at the price of low rate. A particular setting of potential interest in practice is codes of constant rate. For such codes, decoding algo-rithms with locality O(k) were known only for codes of rate Ω(1/), where k is the length of the message. Furthermore, for codes of rate> 1/2, no nontrivial locality had been achieved. In this paper we construct a new family of locally decodable codes that have very efficient local decoding algorithms, and at the same time have rate approaching 1. We show that for every > 0 and α> 0, for infinitely many k, there exists a code C which encodes messages of length k with rate 1 − α, and is locally decodable from a constant fraction of errors using O(k) queries and time. These codes, which we call multiplicity codes, are based on evaluating multivariate polynomials and their derivatives. Multiplicity codes extend traditional multivariate polynomial codes; they inherit the local-decodability of these codes, and at the same time achieve better tradeoffs and flexibility in the rate and minimum distance.