Consider a linear [n, k, d] q code C. We say that that i-th coordinate of C has locality r, if the value at this coordinate can be recovered from accessing some other r coordinates of C. Data storage applications require codes with small redundancy, low locality for information coordinates, large distance, and low locality for parity coordinates. In this paper we carry out an in-depth study of the relations between these parameters.We establish a tight bound for the redundancy n − k in terms of the message length, the distance, and the locality of information coordinates. We refer to codes attaining the bound as optimal. We prove some structure theorems about optimal codes, which are particularly strong for small distances. This gives a fairly complete picture of the tradeoffs between codewords length, worst-case distance and locality of information symbols.We then consider the locality of parity check symbols and erasure correction beyond worst case distance for optimal codes. Using our structure theorem, we obtain a tight bound for the locality of parity symbols possible in such codes for a broad class of parameter settings. We prove that there is a tradeoff between having good locality for parity checks and the ability to correct erasures beyond the minimum distance.

Synthetic DNA is durable and can encode digital data with high density, making it an attractive medium for data storage. However, recovering stored data on a large-scale currently requires all the DNA in a pool to be sequenced, even if only a subset of the information needs to be extracted. Here, we encode and store 35 distinct files (over 200 MB of data), in more than 13 million DNA oligonucleotides, and show that we can recover each file individually and with no errors, using a random access approach. We design and validate a large library of primers that enable individual recovery of all files stored within the DNA. We also develop an algorithm that greatly reduces the sequencing read coverage required for error-free decoding by maximizing information from all sequence reads. These advances demonstrate a viable, large-scale system for DNA data storage and retrieval.

Consider a systematic linear code where some (local) parity symbols depend on few prescribed symbols, while other (heavy) parity symbols may depend on all data symbols. Local parities allow to quickly recover any single symbol when it is erased, while heavy parities provide tolerance to a large number of simultaneous erasures. A code as above is maximally-recoverable, if it corrects all erasure patterns which are information theoretically recoverable given the code topology. In this paper we present explicit families of maximally-recoverable codes with locality. We also initiate the study of the trade-off between maximal recoverability and alphabet size.Definition 3. Let C be a data-local (k, r, h)-code. We say that C is maximally-recoverable if for any set E ⊆ [n], where E is obtained by picking one coordinate from each of k r local groups, puncturing C in coordinates in E yields a maximum distance separable [k + h, k] code.A [k+h, k] MDS code obviously corrects all patterns of h erasures. Therefore a maximally-recoverable data-local (k, r, h)-code corrects all erasure patterns E ⊆ [n] that involve erasing one coordinate per local group, and h additional coordinates. We now argue that any erasure pattern that is not dominated by a pattern above has to be uncorrectable.Lemma 4. Let C be an arbitrary data-local (k, r, h)-code. Let E ⊆ [n] be an erasure pattern. Suppose E affects t local groups and |E| > t + h; then E is not correctable.

Boolean satisfiability problems are an important benchmark for questions about complexity, algorithms, heuristics and threshold phenomena. Recent work on heuristics, and the satisfiability threshold has centered around the structure and connectivity of the solution space. Motivated by this work, we study structural and connectivity-related properties of the space of solutions of Boolean satisfiability problems and establish various dichotomies in Schaefer's framework.On the structural side, we obtain dichotomies for the kinds of subgraphs of the hypercube that can be induced by the solutions of Boolean formulas, as well as for the diameter of the connected components of the solution space. On the computational side, we establish dichotomy theorems for the complexity of the connectivity and stconnectivity questions for the graph of solutions of Boolean formulas. Our results assert that the intractable side of the computational dichotomies is PSPACE-complete, while the tractable side -which includes but is not limited to all problems with polynomial time algorithms for satisfiability -is in P for the st-connectivity question, and in coNP for the connectivity question. The diameter of components can be exponential for the PSPACE-complete cases, whereas in all other cases it is linear; thus, small diameter and tractability of the connectivity problems are remarkably aligned. The crux of our results is an expressibility theorem showing that in the tractable cases, the subgraphs induced by the solution space possess certain good structural properties, whereas in the intractable cases, the subgraphs can be arbitrary.

We address well-studied problems concerning the learnability of parities and halfspaces in the presence of classification noise.Learning of parities under the uniform distribution with random classification noise, also called the noisy parity problem is a famous open problem in computational learning. We reduce a number of basic problems regarding learning under the uniform distribution to learning of noisy parities. We show that under the uniform distribution, learning parities with adversarial classification noise reduces to learning parities with random classification noise. Together with the parity learning algorithm of Blum et al. [5], this gives the first nontrivial algorithm for learning parities with adversarial noise. We show that learning of DNF expressions reduces to learning noisy parities of just logarithmic number of variables. We show that learning of k-juntas reduces to learning noisy parities of k variables. These reductions work even in the presence of random classification noise in the original DNF or junta.We then consider the problem of learning halfspaces over Q n with adversarial noise or finding a halfspace that maximizes the agreement rate with a given set of examples. We prove an essentially optimal hardness factor of 2− , improving the factor of 85 84 − due to Bshouty and Burroughs [8].Finally, we show that majorities of halfspaces are hard to PAC-learn using any representation, based on the cryptographic assumption underlying the Ajtai-Dwork cryptosystem.

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We present the first local list-decoding algorithm for the r th order Reed-Muller code RM(r, m) over F2 for r ≥ 2. Given an oracle for a received word R : F m 2 → F2, our randomized local list-decoding algorithm produces a list containing all degree r polynomials within relative distance (2 −r − ε) from R for any ε > 0 in time poly(m r , ε −r ). The list size could be exponential in m at radius 2 −r , so our bound is optimal in the local setting. Since RM(r, m) has relative distance 2 −r , our algorithm beats the Johnson bound for r ≥ 2. In the setting where we are allowed running-time polynomial in the block-length, we show that list-decoding is possible up to even larger radii, beyond the minimum distance. We give a deterministic list-decoder that works at error rate below J(2 1−r ), where J(δ) denotes the Johnson radius for minimum distance δ. This shows that RM(2, m) codes are list-decodable up to radius η for any constant η < 1 2 in time polynomial in the block-length.Over small fields Fq, we present list-decoding algorithms in both the global and local settings that work up to the list-decoding radius. We conjecture that the list-decoding radius approaches the minimum distance (like over F2), and prove this holds true when the degree is divisible by q − 1.

We present a new approach to constructing unconditional pseudorandom generators against classes of functions that involve computing a linear function of the inputs. We give an explicit construction of a pseudorandom generator that fools the discrete Fourier transforms of linear functions with seed-length that is nearly logarithmic (up to polyloglog factors) in the input size and the desired error parameter. Our result gives a single pseudorandom generator that fools several important classes of tests computable in logspace that have been considered in the literature, including halfspaces (over general domains), modular tests and combinatorial shapes. For all these classes, our generator is the first that achieves near logarithmic seed-length in both the input length and the error parameter. Getting such a seed-length is a natural challenge in its own right, which needs to be overcome in order to derandomize RL -a central question in complexity theory.Our construction combines ideas from a large body of prior work, ranging from a classical construction of [NN93] to the recent gradually increasing independence paradigm of [KMN11, CRSW13, GMR + 12], while also introducing some novel analytic machinery which might find other applications.