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We prove that a random linear code over F q , with probability arbitrarily close to 1, is list decodable at radius 1 − 1/q − with list size L = O(1// 2) and rate R = Ω q (2 /(log 3 (1//))). Up to the polylogarithmic factor in 1// and constant factors depending on q, this matches the lower bound L = Ω q (1// 2) for the list size and upper bound R = O q (2)(More)
We present a new algorithm that produces a well-spaced superset of points conforming to a given input set in any dimension with guaranteed optimal output size. We also provide an approximate Delaunay graph on the output points. Our algorithm runs in expected time O(2<sup>O(d)</sup>(n log n + m)), where n is the input size, m is the output point set size,(More)
We prove a lower estimate on the increase in entropy when two copies of a conditional random variable X|Y , with X supported on Z q = {0, 1,. .. , q − 1} for prime q, are summed modulo q. Specifically, given two i.i.d. copies (X 1 , Y 1) and (X 2 , Y 2) of a pair of random variables (X, Y), with X taking values in Z q , we show H(X 1 + X 2 | Y 1 , Y 2) −(More)
Locally testable codes (LTCs) of constant minimum (absolute) distance that allow the tester to make a nearly linear number of queries have become the focus of attention recently, due to their connections to central questions in approximability theory. In particular, the binary Reed-Muller code of block length N and absolute distance d is known to be(More)
Various combinatorial/algebraic parameters are used to quantify the complexity of a Boolean function. Among them, sensitivity is one of the simplest and block sensitivity is one of the most useful. Nisan (1989) and Nisan and Szegedy (1991) showed that block sensitivity and several other parameters, such as certificate complexity, decision tree depth, and(More)
Several researchers proposed using non-Euclidean metrics on point sets in Euclidean space for clustering noisy data. Almost always, a distance function is desired that recognizes the closeness of the points in the same cluster, even if the Euclidean cluster diameter is large. Therefore , it is preferred to assign smaller costs to the paths that stay close(More)
We study the communication rate of coding schemes for interactive communication that transform any two-party interactive protocol into a protocol that is robust to noise. Recently, Haeupler [Hae14] showed that if an > 0 fraction of transmissions are corrupted, adversarially or randomly, then it is possible to achieve a communication rate of 1 − O(√).(More)
We introduce the notion of one-way communication schemes with partial noiseless feedback. In this setting, Alice wishes to communicate a message to Bob by using a communication scheme that involves sending a sequence of bits over a channel while receiving feedback bits from Bob for δ fraction of the transmissions. An adversary is allowed to corrupt up to a(More)
In a breakthrough work, Marcus et al. [15] recently showed that every d-regular bipartite Ramanujan graph has a 2-lift that is also d-regular bipartite Ramanujan. As a consequence, a straightforward iterative brute-force search algorithm leads to the construction of a d-regular bipartite Ramanujan graph on N vertices in time 2 O(dN). Shift k-lifts studied(More)
  • Curtis Mcmullen, Yum-Tong Siu, Emily Riehl, Ameya Velingker, Alexandra Kolla, Jonathan Kelner +3 others
  • 2015
Acknowledgements First, I'd like to thank my family for their love and encouragement, without which I wouldn't have been able to follow my passion for mathematics. Any success I've had is rooted in the eeort they invested in my upbringing. Next, this thesis would not have been possible without the support of my advisor, Professor Salil Vadhan, who pointed(More)