For any odd integer q > 1, we improve the lower bound for general q-query locally decodablecodes C : {0, 1}n → {0, 1}m from m = Ω(n/ logn)q+1q−1 to m = Ω( nq+1q−1) / logn. For example,for q = 3 we improve the previous bound from Ω(n/ log n) to Ω(n/ logn). For linear 3-querylocally decodable codes C : F → F, we improve the lower bound further to Ω(n/ log log n...

A Private Information Retrieval (PIR) protocol enables a user to retrieve a data item from a database while hiding the identity of the item being retrieved. In a t-private, k-server PIR protocol the database is replicated among k servers, and the user’s privacy is protected from any collusion of up to t servers. The main cost-measure of such protocols is the communication complexity of retrievi...

Let C : {0, 1}n 7→ {0, 1}m be a code encoding an n-bit string into an m-bit string. Such a code is called a (q, c, ǫ) smooth code if there exists a decoding algorithm which while decoding any bit of the input, makes at most q probes on the code word and the probability that it looks at any location is at most c/m. The error made by the decoding algorithm is at most ǫ. Smooth codes were introduc...

The US Associated Pacific Island Nations (USAPIN) are developing cancer data and surveillance infrastructure as a component of their comprehensive cancer control plans. Various cancer registry systems were assessed for adaptability, sustainability, and technical feasibility in resource limited, relatively small island populations. Many jurisdictions lack systematic cancer data collection or cod...

We study Locally Testable Codes (LTCs) that can be tested by making two queries to the tested word using an a ne test. That is, we consider LTCs over a nite eld F, with codeword testers that only use tests of the form avi + bvj = c, where v is the tested word and a, b, c ∈ F. We show that such LTCs, with high minimal distance, must be of constant size. Speci cally, we show that every 2-query LT...

We study Locally Testable Codes (LTCs) that can be tested by making two queries to the tested word using an a ne test. That is, we consider LTCs over a nite eld F, with codeword testers that only use tests of the form avi + bvj = c, where v is the tested word and a, b, c ∈ F. We show that such LTCs, with high minimal distance, must be of constant size. Speci cally, we show that every 2-query LT...

Motivated by two problems on arithmetic progressions (APs)—concerning large deviations for AP counts in random sets and random differences in Szemerédi’s theorem— we prove upper bounds on the Gaussian width of the image of the n-dimensional Boolean hypercube under a mapping ψ : R → R, where each coordinate is a constant-degree multilinear polynomial with 0/1 coefficients. We show the following ...

We survey the state of the art in constructions of locally testable codes, locally decodable codes and locally correctable codes of high rate.