Strong LTCs with Inverse Poly-Log Rate and Constant Soundness

Viderman, Michael

doi:10.1109/focs.2013.43

Cited by 13 publications

(9 citation statements)

References 59 publications

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“…Although recently some progress was made towards showing the impossibility of such linear length LTCs [5,3], there are known constructions of strong-LTCs with relatively good parameters: Goldreich and Sudan [13] constructed a strong-LTC with constant relative distance and nearly-linear length, where throughout this paper a code of dimension k is said to have nearly-linear length if its codewords are of length k 1+α for an arbitrarily small constant α > 0. Furthermore, recently Viderman [23] constructed a strong-LTC with constant relative distance and quasilinear length (i.e., length k • polylogk).…”

Section: Introductionmentioning

confidence: 99%

Strong Locally Testable Codes with Relaxed Local Decoders

Goldreich

Gur

Komargodski

2019

ACM Trans. Comput. Theory

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Locally testable codes (LTCs) are error-correcting codes that admit very efficient codeword tests. An LTC is said to be strong if it has a proximity-oblivious tester; that is, a tester that makes only a constant number of queries and reject non-codewords with probability that depends solely on their distance from the code. Locally decodable codes (LDCs) are complimentary to LTCs. While the latter allow for highly efficient rejection of strings that are far from being codewords, LDCs allow for highly efficient recovery of individual bits of the information that is encoded in strings that are close to being codewords. Constructions of strong-LTCs with nearly-linear length are known, but the existence of a constant-query LDC with polynomial length is a major open problem. In an attempt to bypass this barrier, Ben-Sasson et al. (SICOMP 2006) introduced a natural relaxation of local decodability, called relaxed-LDCs. This notion requires local recovery of nearly all individual informationbits, yet allows for recovery-failure (but not error) on the rest. Ben-Sasson et al. constructed a constant-query relaxed-LDC with nearly-linear length (i.e., length k 1+α for an arbitrarily small constant α > 0, where k is the dimension of the code). This work focuses on obtaining strong testability and relaxed decodability simultaneously. We construct a family of binary linear codes of nearly-linear length that are both strong-LTCs (with one-sided error) and constant-query relaxed-LDCs. This improves upon the previously known constructions, which either obtain weak LTCs or require polynomial length. Our construction heavily relies on tensor codes and PCPs. In particular, we provide strong canonical PCPs of proximity for membership in any linear code with constant rate and relative distance. Loosely speaking, these are PCPs of proximity wherein the verifier is proximity oblivious (similarly to strong-LTCs) and every valid statement has a unique canonical proof. Furthermore, the verifier is required to reject non-canonical proofs (even for valid statements). As an application, we improve the best known separation result between the complexity of decision and verification in the setting of property testing.

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Section: Introductionmentioning

confidence: 99%

Strong Locally Testable Codes with Relaxed Local Decoders

Goldreich

Gur

Komargodski

2019

ACM Trans. Comput. Theory

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“…Another related notion is that of locally testable codes [GS06], which are, loosely speaking, codes for which there exist a probabilistic algorithm that accepts valid codewords, and rejects inputs that are "far" in Hamming distance from any codeword, while only probing a small fraction of the input. Much stronger upper bounds are known for locally testable codes than for LDCs, and in particular, there exists O(1)-local LTCs with blocklength n = k • polylog(k) [GS06] (see also [Mei09,Vid13]). It is also known that LDCs do not imply locally testable codes and vice versa [KV10].…”

Section: Related Workmentioning

confidence: 99%

On the Power of Relaxed Local Decoding Algorithms

Gur¹,

Lachish²

2021

SIAM J. Comput.

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“…The starting point for this paper is the recent result of [23] on high-rate list recoverable tensor codes, and its corollaries. Tensoring is a natural operation on codes that significantly enhances their local properties [5,34,9,10,15,6,7,37,28,36,26].…”

Section: The Contextmentioning

confidence: 99%

On List Recovery of High-Rate Tensor Codes

Kopparty

Resch

Ron-Zewi

et al. 2021

IEEE Trans. Inform. Theory

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We continue the study of list recovery properties of high-rate tensor codes, initiated by Hemenway, Ron-Zewi, and Wootters (FOCS'17). In that work it was shown that the tensor product of an efficient (poly-time) high-rate globally list recoverable code is approximately locally list recoverable, as well as globally list recoverable in probabilistic near-linear time. This was used in turn to give the first capacity-achieving list decodable codes with (1) local list decoding algorithms, and with (2) probabilistic near-linear time global list decoding algorithms. This also yielded constant-rate codes approaching the Gilbert-Varshamov bound with probabilistic near-linear time global unique decoding algorithms. In the current work we obtain the following results: 1. The tensor product of an efficient (poly-time) high-rate globally list recoverable code is globally list recoverable in deterministic near-linear time. This yields in turn the first capacity-achieving list decodable codes with deterministic near-linear time global list decoding algorithms. It also gives constant-rate codes approaching the Gilbert-Varshamov bound with deterministic near-linear time global unique decoding algorithms. 2. If the base code is additionally locally correctable, then the tensor product is (genuinely) locally list recoverable. This yields in turn (non-explicit) constant-rate codes approaching the Gilbert-Varshamov bound that are locally correctable with query complexity and running time N o(1). This improves over prior work by Gopi et. al. (SODA'17; IEEE Transactions on Information Theory'18) that only gave query complexity N ε with rate that is exponentially small in 1/ε. 3. A nearly-tight combinatorial lower bound on output list size for list recovering high-rate tensor codes. This bound implies in turn a nearly-tight lower bound of N Ω(1/ log log N) on the product of query complexity and output list size for locally list recovering high-rate tensor codes. 2012 ACM Subject Classification Mathematics of computing → Coding theory; Theory of computation → Pseudorandomness and derandomization Keywords and phrases Coding theory, Tensor codes, List-decoding and recovery, Local codes

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Strong LTCs with Inverse Poly-Log Rate and Constant Soundness

Cited by 13 publications

References 59 publications

Strong Locally Testable Codes with Relaxed Local Decoders

Strong Locally Testable Codes with Relaxed Local Decoders

On the Power of Relaxed Local Decoding Algorithms

On List Recovery of High-Rate Tensor Codes

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