Locally Testable Codes and Cayley Graphs

We give two new characterizations of (\F_2-linear) locally testable error-correcting codes in terms of Cayley graphs over \F_2^h: \begin{enumerate} \item A locally testable code is equivalent to a Cayley graph over \F_2^h whose set of generators is significantly larger than $h$ and has no short linear dependencies, but yields a shortest-path metric that embeds into $\ell_1$ with constant distortion. This extends and gives a converse to a result of Khot and Naor (2006), which showed that codes with large dual distance imply Cayley graphs that have no low-distortion embeddings into $\ell_1$. \item A locally testable code is equivalent to a Cayley graph over \F_2^h that has significantly more than $h$ eigenvalues near 1, which have no short linear dependencies among them and which "explain" all of the large eigenvalues. This extends and gives a converse to a recent construction of Barak et al. (2012), which showed that locally testable codes imply Cayley graphs that are small-set expanders but have many large eigenvalues. \end{enumerate}Comment: 22 page

Subspace polynomials and list decoding of Reed-Solomon codes

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, February 2007.Includes bibliographical references (p. 29-31).We show combinatorial limitations on efficient list decoding of Reed-Solomon codes beyond the Johnson and Guruswami-Sudan bounds [Joh62, Joh63, GS99]. In particular, we show that for any ... , there exist arbitrarily large fields ... * Existence: there exists a received word ... that agrees with a super-polynomial number of distinct degree K polynomials on ... points each; * Explicit: there exists a polynomial time constructible received word ... that agrees with a super-polynomial number of distinct degree K polynomials, on ... points each. Ill both cases, our results improve upon the previous state of the art, which was , NM/6 for the existence case [JH01], and a ... for the explicit one [GR,05b]. Furthermore, for 6 close to 1 our bound approaches the Guruswami-Sudan bound (which is ... ) and rules out the possibility of extending their efficient RS list decoding algorithm to any significantly larger decoding radius. Our proof method is surprisingly simple. We work with polynomials that vanish on subspaces of an extension field viewed as a vector space over the base field.(cont.) These subspace polynomials are a subclass of linearized polynomials that were studied by Ore [Ore33, Ore34] in the 1930s and by coding theorists. For us their main attraction is their sparsity and abundance of roots. We also complement our negative results by giving a list decoding algorithm for linearized polynomials beyond the Johnson-Guruswami-Sudan bounds.by Swastik Kopparty.S.M

A PCP Characterization of AM

We introduce a 2-round stochastic constraint-satisfaction problem, and show that its approximation version is complete for (the promise version of) the complexity class AM. This gives a `PCP characterization' of AM analogous to the PCP Theorem for NP. Similar characterizations have been given for higher levels of the Polynomial Hierarchy, and for PSPACE; however, we suggest that the result for AM might be of particular significance for attempts to derandomize this class. To test this notion, we pose some `Randomized Optimization Hypotheses' related to our stochastic CSPs that (in light of our result) would imply collapse results for AM. Unfortunately, the hypotheses appear over-strong, and we present evidence against them. In the process we show that, if some language in NP is hard-on-average against circuits of size 2^{Omega(n)}, then there exist hard-on-average optimization problems of a particularly elegant form. All our proofs use a powerful form of PCPs known as Probabilistically Checkable Proofs of Proximity, and demonstrate their versatility. We also use known results on randomness-efficient soundness- and hardness-amplification. In particular, we make essential use of the Impagliazzo-Wigderson generator; our analysis relies on a recent Chernoff-type theorem for expander walks.Comment: 18 page

Efficient and Error-Correcting Data Structures for Membership and Polynomial Evaluation

We construct efficient data structures that are resilient against a constant fraction of adversarial noise. Our model requires that the decoder answers most queries correctly with high probability and for the remaining queries, the decoder with high probability either answers correctly or declares "don't know." Furthermore, if there is no noise on the data structure, it answers all queries correctly with high probability. Our model is the common generalization of a model proposed recently by de Wolf and the notion of "relaxed locally decodable codes" developed in the PCP literature. We measure the efficiency of a data structure in terms of its length, measured by the number of bits in its representation, and query-answering time, measured by the number of bit-probes to the (possibly corrupted) representation. In this work, we study two data structure problems: membership and polynomial evaluation. We show that these two problems have constructions that are simultaneously efficient and error-correcting.Comment: An abridged version of this paper appears in STACS 201

Knowledge implies time/space efficiency

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2007.Includes bibliographical references (p. 35-36).The probabilistically checkable proof (PCP) system enables proofs to be verified in time polylogarithmic in the length of a classical proof. Computationally sound (CS) proofs improve upon PCPs by additionally shortening the length of the transmitted proof to be polylogarithmic in the length of the classical proof. In this thesis we explore the ultimate limits of non-interactive proof systems with respect to time/space efficiency and the new criterion of composability. We deduce the existence of our proposed proof system by way of a natural new assumption about proofs of knowledge. In fact, a main contribution of our result is showing that knowledge can be "traded" for time and space efficiency in noninteractive proof systems.by Paul Valiant.S.M