NDLI: Towards Lower Bounds on Locally Testable Codes via Density Arguments

Content Provider	IEEE Xplore Digital Library
Author	Ben-Sasson, E. Viderman, M.
Copyright Year	2011
Abstract	The main open problem in the area of locally testable codes (LTCs) is whether there exists an asymptotically good family of LTCs and to resolve this question it suffices to consider the case of query complexity $3$. We argue that to refute the existence of such an asymptotically good family one should prove that the number of dual codewords of weight at most $3$ is super-linear in the blocklength of the code. The main technical contribution of this paper is an improvement of the combinatorial lemma of Goldreich et al. [2006] which bounds the rate of $2$-query locally decodable codes (LDCs) and is used in state-of-the-art rate-bounds for linear LDCs. The lemma of Goldreich et al. bounds the rate of $2$-query LDCs of blocklength $n$ in terms of the corruption parameter $\delta(n)$ --- this is the maximal fraction of corrupted codeword bits for which a ($2$-query) decoder can recover correctly every message bit (with high probability). Our combinatorial lemma gives nontrivial rate bounds for any corruption parameter $\delta(n)$ such that $\delta(n) \cdot n =\omega(1)$, whereas the previous lemma works only for corruption parameter $\delta(n)$ such that $\delta(n) \cdot n \geq \log n$. The study of LDCs with sublinear corruption parameter is also motivated by Dvir's [2010] observation that sufficiently strong bounds on the rate of such LDCs imply explicit constructions of rigid matrices.
Starting Page	66
Ending Page	76
File Size	288479
Page Count	11
File Format	PDF
ISBN	9781457701795
ISSN	10930159
DOI	10.1109/CCC.2011.9
Language	English
Publisher	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Publisher Date	2011-06-08
Publisher Place	USA
Access Restriction	Subscribed
Rights Holder	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subject Keyword	Parity check codes Linear code Complexity theory Linear matrix inequalities Decoding Upper bound Error correction codes lower bounds linear codes locally testable codes locally decodable codes LDPC codes
Content Type	Text
Resource Type	Article

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