Error correcting codes encode messages by codewords in such a way that even if some of the codeword is corrupted, the message can be decoded. Typical decoding algorithms for error correcting codes either use linear space or quadratic time. A natural question is whether codes can be decoded in near-linear time and sub-linear space simultaneously. A recent result by Cook and Moshkovitz gave efficient decoders that can uniquely decode Reed-Muller and other codes from a constant fraction (less than half) of corruption. In this work, we address the problem of list decoding in near-linear time and sub-linear space. In the list decoding setting, most of the codeword is corrupted, and one wants to output a short list of potential messages that contains the true message. For any constants gamma, Tau > 0, we give decoders for Reed-Muller codes that can decode from 1-gamma fraction of corruptions in time n(1+t) and space n(t). Our decoders work by extending the iterative correction technique of Cook and Moshkovitz. However, that technique, which gradually decreases the number of corruptions in the message, was tailored to the unique decoding setting. We first identify an intermediate problem, codewords list recovery, for which we can make iterative correction work. We then show how to reduce general list decoding to the codewords list recovery problem in efficient time and space. The reduction relies on local correction and testing. In the codewords list recovery problem, the input consists of n unordered lists containing exactly the symbols from L codewords, where a small fraction of the lists is corrupted. The goal is to find the L codewords. In addition, we prove that any linear code with time-space efficient encoding or decoding must be local, in the sense that the codewords satisfy a local linear constraint. This rules out codes like Reed-Solomon from having time-space efficient encoding or decoding. 2012 ACM Subject Classification Theory of computation. Error-correcting codes; Theory of computation -> Pseudorandomness and derandomization

Time efficient decoding algorithms for error correcting codes often require linear space. However, locally decodable codes yield more efficient randomized decoders that run in time n(1+o(1)) and space n(o(1)). In this work we focus on deterministic decoding. Gronemeier showed that any non-adaptive deterministic decoder for a good code running in time n(1+delta) must use space n(1-delta). In sharp contrast, we show that all typical locally correctable codes have (non-uniform) time and space efficient adaptive deterministic decoders. To obtain the decoders, we devise a new timespace efficient derandomization technique that works by iterative correction. Further, we give a new construction of curve samplers that allow us to uniformly decode Reed-Muller codes time and space efficiently. In particular, for any constant gamma > 0, we give asymptotically good Reed-Muller codes that are decodable in time n(1+gamma). and space n(gamma). by a uniform, deterministic decoder. A related construction allows us to uniformly decode asymptotically good codes based on lifted Reed-Solomon codes in time n(1+o(1)) and space n(o(1)).

This paper studies expansion properties of the (generalized) Johnson Graph. For natural numbers t < ` < k, the nodes of the graph are sets of size ` in a universe of size k. Two sets are connected if their intersection is of size t. The Johnson graph arises often in combinatorics and theoretical computer science: it represents a “slice” of the noisy hypercube, and it is the graph that underlies direct product tests as well as a candidate hard unique game. We prove that any small set of vertices in the graph either has near perfect edge expansion or is not pseudorandom. Here “not pseudorandom” means that the set becomes denser when conditioning on containing a small set of elements. In other words, we show that slices of the noisy hypercube – while not small set expanders like the noisy hypercube – only have non-expanding small sets of a certain simple structure. This paper is related to a recent line of work establishing the 2-to-2 Theorem in PCP. The result was motivated, in part, by [7] which hypothesized and made partial progress on similar result for the Grassmann graph. In turn, our result for the Johnson graphs served as a crucial step towards the full result for the Grassmann graphs completed subsequently in [20].

A natural model of a source of randomness consists of a long stream of symbols X = X-1 omicron...omicron X-t, with some guarantee on the entropy of X-i conditioned on the outcome of the prefix x(1),..., x(i-1). We study unpredictable sources, a generalization of the almost Chor-Goldreich (CG) sources considered in [9]. In an unpredictable source X, for a typical draw of x similar to X, for most i-s, the element x(i) has a low probability of occurring given x(1),..., x(i-1). Such a model relaxes the often unrealistic assumption of a CG source that for every i, and every x(1),..., x(i-1), the next symbol X-i has sufficiently large entropy. Unpredictable sources subsume all previously considered notions of almost CG sources, including notions that [9] failed to analyze, and including those that are equivalent to general sources with high min entropy. For a lossless expander G = ( V, E) with m = log |V|, we consider a random walk V-0, V-1,..., V-t on G using unpredictable instructions that have sufficient entropy with respect to m. Our main theorem is that for almost all the steps t/2 <= i <= t in the walk, the vertex V-i is close to a distribution with min-entropy at least m - O(1). As a result, we obtain seeded online condensers with constant entropy gap, and seedless (deterministic) condensers outputting a constant fraction of the entropy. In particular, our condensers run in space comparable to the output entropy, as opposed to the size of the stream, and even when the length t of the stream is not known ahead of time. As another corollary, we obtain a new extractor based on expander random walks handling lower entropy than the classic expander based construction relying on spectral techniques [11]. As our main technical tool, we provide a novel analysis covering a key case of adversarial random walks on lossless expanders that [9] fails to address. As part of the analysis, we provide a "chain rule for vertex probabilities". The standard chain rule states that for every x similar to X and i, Pr( x(1),..., x(i)) = Pr[ X-i = x(i)|X-[1,X- i-1] - x(1),..., x(i-1)] center dot Pr( x(1),..., x(i-1)). If W( x(1),..., x(i)) is the vertex reached using x(1),..., x(i), then the chain rule for vertex probabilities essentially states that the same phenomena occurs for a typical x: Pr[V-i = W( x(1),..., x(i))]less than or similar to Pr[ X-i = x(i)|X-[1,X- i-1] = x(1),..., xi-1] center dot Pr[ Vi-1 = W( x(1),..., x(i-1))], where V-i is the vertex distribution of the random walk at step i using X.

We give the first explicit constant rate, constant relative distance, linear codes with an encoder that runs in time n(1) (+) (o(1)) and space polylog(n) provided random access to the message. Prior to this work, the only such codes were non-explicit, for instance repeat accumulate codes [19] and the codes described in [26]. To construct our codes, we also give explicit, efficiently invertible, lossless condensers with constant entropy gap and polylogarithmic seed length. In contrast to encoders with random access to the message, we show that encoders with sequential access to the message can not run in almost linear time and polylogarithmic space. Our notion of sequential access is much stronger than streaming access.

We present simple constructions of good approxi-mate locally decodable codes (ALDCs) in the presence of a $\delta{-}$ fraction of errors for $\delta < 1/2$ . In a standard locally decodable code $C:\Sigma_{1}^{k}\rightarrow\Sigma_{2}^{n}$ , there is a decoder $M$ that on input $i\in[k]$ correctly outputs the i-th symbol of a message $x$ (with high probability) using only $q$ queries to a given string $w$ that is $\Delta$ -close to $C(x)$ . In an ALDC, the decoder $M$ only needs to be correct on a 1 $-\varepsilon$ fraction of $i\in[k]$ for $\varepsilon$ much smaller than $\delta$ . We present a construction of explicit ALDCs for all constants 1/2 $> \delta > \varepsilon$ with a constant number of queries $q$ and with constant, near-optimal rate. Standard LDCs with constant number of queries and any constant rate are known to be impossible. We additionally explore what is the lowest error probability $\varepsilon$ one can achieve for fixed $\delta$ and $q$ . We show that for any ALDC, $\in=\Omega(\delta \mathrm{r}q/2\rceil)$ . We then show that there exist explicit constant rate ALDCs for any constant $q$ that achieve $\varepsilon=O(\delta^{\lceil q/2\rceil})$ . In particular, for $q=3$ , we have a constant rate ALDC with error probability $\varepsilon=O(\delta^{2})$ . A full version of this paper is available at https://eccc.weizmann.ac.il/report/2023/056/.

In a regular PCP the verifier queries each proof symbol in the same number of tests. This number is called the degree of the proof, and it is at least 1/(sq) where s is the soundness error and q is the number of queries. It is incredibly useful to have regularity and reduced degree in PCP. There is an expander-based transformation by Papadimitriou and Yannakakis that transforms any PCP with a constant number of queries and constant soundness error to a regular PCP with constant degree. There are also transformations for low error projection and unique PCPs. Other PCPs are constructed especially to be regular. In this work we show how to regularize and reduce degree of PCPs with a possibly large number of queries and low soundness error. As an application, we prove NP-hardness of an unweighted variant of the collective minimum monotone satisfying assignment problem, which was introduced by Hirahara (FOCS'22) to prove NP-hardness of MCSP* (the partial function variant of the Minimum Circuit Size Problem) under randomized reductions. We present a simplified proof and sufficient conditions under which MCSP* is NP-hard under the standard notion of reduction: MCSP* is NP-hard under deterministic polynomial-time many-one reductions if there exists a function in E that satisfies certain direct sum properties.

We prove that for some constant a > 1, for all k ≤ a , MATIME [ n k + o (1) ] / 1 ̸⊂ SIZE [ O ( n k )] , for some specific o (1) function. This is a super linear polynomial circuit lower bound. Previously, Santhanam [29] showed that there exists a constant c > 1 such that for all k > 1: MATIME [ n ck ] / 1 ̸⊂ SIZE [ O ( n k )] . Inherently to Santhanam’s proof, c is a large constant and there is no upper bound on c . Using ideas from Murray and Williams [26], one can show for all k > 1: MATIME [ n 10 k 2 ] / 1 ̸⊂ SIZE [ O ( n k )] . To prove this result, we construct the first PCP for SPACE [ n ] with quasi-linear verifier time: our PCP has a ˜ O ( n ) time verifier, ˜ O ( n ) space prover, O (log( n )) queries, and polynomial alphabet size. Prior to this work, PCP s for SPACE [ O ( n )] had verifiers that run in Ω( n 2 ) time. This PCP also proves that NE has MIP verifiers which run in time ˜ O ( n ).

A Chor–Goldreich (CG) source is a sequence of random variables X = X 1 ∘ … ∘ X t , where each X i ∼ {0,1} d and X i has δ d min-entropy conditioned on any fixing of X 1 ∘ … ∘ X i −1 . The parameter 0<δ≤ 1 is the entropy rate of the source. We typically think of d as constant and t as growing. We extend this notion in several ways, defining almost CG sources. Most notably, we allow each X i to only have conditional Shannon entropy δ d . We achieve pseudorandomness results for almost CG sources which were not known to hold even for standard CG sources, and even for the weaker model of Santha–Vazirani sources: We construct a deterministic condenser that on input X , outputs a distribution which is close to having constant entropy gap, namely a distribution Z ∼ {0,1} m for m ≈ δ dt with min-entropy m − O (1). Therefore, we can simulate any randomized algorithm with small failure probability using almost CG sources with no multiplicative slowdown. This result extends to randomized protocols as well, and any setting in which we cannot simply cycle over all seeds, and a “one-shot” simulation is needed. Moreover, our construction works in an online manner, since it is based on random walks on expanders. Our main technical contribution is a novel analysis of random walks, which should be of independent interest. We analyze walks with adversarially correlated steps, each step being entropy-deficient, on good enough lossless expanders. We prove that such walks (or certain interleaved walks on two expanders), starting from a fixed vertex and walking according to X 1 ∘ … ∘ X t , accumulate most of the entropy in X .

Existing proofs that deduce BPP = P from circuit lower bounds convert randomized algorithms into deterministic algorithms with a large polynomial slowdown. We convert randomized algorithms into deterministic ones with little slowdown. Specifically, assuming exponential lower bounds against randomized single-valued nondeterministic (SVN) circuits, we convert any randomized algorithm over inputs of length n running in time t ≥ n to a deterministic one running in time t 2+α for an arbitrarily small constant . Such a slowdown is nearly optimal, as, under complexity-theoretic assumptions, there are problems with an inherent quadratic derandomization slowdown. We also convert any randomized algorithm that errs rarely into a deterministic algorithm having a similar running time (with pre-processing). The latter derandomization result holds under weaker assumptions, of exponential lower bounds against deterministic SVN circuits. Our results follow from a new, nearly optimal, explicit pseudorandom generator fooling circuits of size s with seed length (1 + α)log s, under the assumption that there exists a function f ϵ E that requires randomized SVN circuits of size at least 2 (1-α')n , where. α=O(α'). The construction uses, among other ideas, a new connection between pseudoentropy generators and locally list recoverable codes.

We reduce the problem of proving a "Boolean Unique Games Conjecture" (with gap 1-delta vs. 1-C*delta, for any C> 1, and sufficiently small delta>0) to the problem of proving a PCP Theorem for a certain non-unique game. In a previous work, Khot and Moshkovitz suggested an inefficient candidate reduction (i.e., without a proof of soundness). The current work is the first to provide an efficient reduction along with a proof of soundness. The non-unique game we reduce from is similar to non-unique games for which PCP theorems are known. Our proof relies on a new concentration theorem for functions in Gaussian space that are restricted to a random hyperplane. We bound the typical Euclidean distance between the low degree part of the restriction of the function to the hyperplane and the restriction to the hyperplane of the low degree part of the function.

We give an efficient deterministic algorithm that outputs an expanding generating set for any finite abelian group. The size of the generating set is close to the randomized construction of Alon and Roichman (1994), improving upon various deterministic constructions in both the dependence on the dimension and the spectral gap. By obtaining optimal dependence on the dimension we resolve a conjecture of Azar, Motwani, and Naor (1998) in the affirmative. Our technique is an extension of the bias amplification technique of Ta-Shma (2017), who used random walks on expanders to obtain expanding generating sets over the additive group of n-bit strings. As a consequence, we obtain (i) randomness-efficient constructions of almost k-wise independent variables, (ii) a faster deterministic algorithm for the Remote Point Problem, (iii) randomness-efficient low-degree tests, and (iv) randomness-efficient verification of matrix multiplication.

Strong Parallel Repetition for Unique Games on Small Set Expanders The strong parallel repetition problem for unique games is to efficiently reduce the 1-delta vs. 1-C*delta gap problem of Boolean unique games (where C>1 is a sufficiently large constant) to the 1-epsilon vs. epsilon gap problem of unique games over large alphabet. Due to its importance to the Unique Games Conjecture, this problem garnered a great deal of interest from the research community. There are positive results for certain easy unique games (e.g., unique games on expanders), and an impossibility result for hard unique games. In this paper we show how to bypass the impossibility result by enlarging the alphabet sufficiently before repetition. We consider the case of unique games on small set expanders for two setups: (i) Strong small set expanders that yield easy unique games. (ii) Weaker small set expanders underlying possibly hard unique games as long as the game is mildly fortified. We show how to fortify unique games in both cases, i.e., how to transform the game so sufficiently large induced sub-games have bounded value. We then prove strong parallel repetition for the fortified games. Prior to this work fortification was known for projection games but seemed hopeless for unique games.

We present a randomness efficient version of the linear noise operator Tρ from boolean function analysis by constructing a sparse linear operator on the space of boolean functions {0, 1} → {0, 1} with similar eigenvalue profile to Tρ. The linear operator we construct is a direct consequence of a generalization of -biased sets to the product distribution Dp on {0, 1} where the marginal of each coordinate is p = 2 − 1 2ρ. Such a generalization is a small support distribution that fools linear tests when the input of the test comes from Dp instead of the uniform distribution. We give an explicit construction of such a distribution that requires logn+Op(log logn+ log 1 ) bits of uniform randomness to sample from, where the p subscript hides O(log2 1 p ) factors. When p and are constant, this yields a support size nearly linear in n, whereas previous best known constructions only guarantee a size of poly(n). Furthermore, our construction implies an explicitly constructible “sparse” noisy hypercube graph that is a small set expander. 2012 ACM Subject Classification Theory of computation → Pseudorandomness and derandomization

We present techniques for decreasing the error probability of randomized algorithms and for converting randomized algorithms to deterministic (nonuniform) algorithms. Unlike most existing techniques that involve repetition of the randomized algorithm and hence a slowdown, our techniques produce algorithms with a similar run-time to the original randomized algorithms. The amplification technique is related to a certain stochastic multi-armed bandit problem. The derandomization technique - which is the main contribution of this work - points to an intriguing connection between derandomization and sketching/sparsification. We demonstrate the techniques by showing algorithms for approximating free games (constraint satisfaction problems on dense bipartite graphs).

The PCP (i.e., Probabilistically Checkable Proofs) Theorem [8, 7, 22, 5, 4] states that any mathematical proof can be converted to a format that can be checked by a veri er making only a constant number of queries to the proof. The veri er picks the queries in a randomized way and might err with low probability.

With any hypothesis class one can associate a bipartite graph whose vertices are the hypotheses H on one side and all possible labeled examples X on the other side, and an hypothesis is connected to all the labeled examples that are consistent with it. We call this graph the hypotheses graph. We prove that any hypothesis class whose hypotheses graph is mixing cannot be learned using less than Omega(log^2 |H|) memory bits unless the learner uses at least a large number |H|^Omega(1) labeled examples. Our work builds on a combinatorial framework that we suggested in a previous work for proving lower bounds on space bounded learning. The strong lower bound is obtained by defining a new notion of pseudorandomness, the entropy sampler. Raz obtained a similar result using different ideas.

With any hypothesis class one can associate a bipartite graph whose vertices are the hypotheses H on one side and all possible labeled examples X on the other side, and an hypothesis is connected to all the labeled examples that are consistent with it. We call this graph the hypotheses graph. We prove that any hypothesis class whose hypotheses graph is mixing cannot be learned using less than 2 2 |H|) memory states unless the learner uses at least a large number of |H| labeled examples. In contrast, there is a learner that uses 2Θ(log|X | log |H|) memory states and only Θ(log |H|) labeled examples, and there is a learner that uses only |H| memory states but a large number Θ(|H| log |H|) of labeled examples. Our work builds on a combinatorial framework we suggested in a previous work for proving lower bounds on space bounded learning. The strong lower bound is obtained by considering a new notion of pseudorandomness for a sequence of graphs that represents the learner. ∗danama@cs.utexas.edu. Department of Computer Science, UT Austin. This material is based upon work supported by the National Science Foundation under grants number 1218547 and 1648712. †michal.moshkovitz@mail.huji.ac.il. The Edmond and Lily Safra Center for Brain Sciences, Hebrew University. This work is partially supported by the Gatsby Charitable Foundation, The Israel Science Foundation, and Intel ICRI-CI center. M.M. is grateful to the Harry and Sylvia Hoffman Leadership and Responsibility Program. ISSN 1433-8092 Electronic Colloquium on Computational Complexity, Report No. 116 (2017)

Many known optimal NP-hardness of approximation results are reductions from a problem called L abel -C over . The input is a bipartite graph G = (L, R, E) and each edge e = (x, y) ∈ E carries a projection πe that maps labels to x to labels to y. The objective is to find a labeling of the vertices that satisfies as many of the projections as possible. It is believed that the best approximation ratio efficiently achievable for L abel -C over is of the form N−c where N = nk, n is the number of vertices, k is the number of labels, and 0 Inspired by a framework originally developed for D ensest k-S ubgraph , we propose a "log density threshold" for the approximability of Label-Cover. Specifically, we suggest the possibility that the Label-Cover approximation problem undergoes a computational phase transition at the same threshold at which local algorithms for its random counterpart fail. This threshold is [EQUATION]. We then design, for any e > 0, a polynomial-time approximation algorithm for semi-random L abel -C over whose approximation ratio is [EQUATION]. In our semi-random model, the input graph is random (or even just expanding), and the projections on the edges are arbitrary. For worst-case L abel -C over we show a polynomial-time algorithm whose approximation ratio is roughly N−0.233. The previous best efficient approximation ratio was N−0.25. We present some evidence towards an N−c threshold by constructing integrality gaps for NΩ(1) rounds of the Sum-of-squares/Lasserre hierarchy of the natural relaxation of Label Cover. For general 2CSP the "log density threshold" is N−0.25, and we give a polynomial-time algorithm in the semi-random model whose approximation ratio is N−0.25+e for any e > 0.