Deterministic Extractors for Additive Sources

We propose a new model of a weakly random source that admits randomness extraction. Our model of additive sources includes such natural sources as uniform distributions on arithmetic progressions (APs), generalized arithmetic progressions (GAPs), and Bohr sets, each of which generalizes affine sources. We give an explicit extractor for additive sources with linear min-entropy over both $\mathbb{Z}_p$ and $\mathbb{Z}_p^n$, for large prime $p$, although our results over $\mathbb{Z}_p^n$ require that the source further satisfy a list-decodability condition. As a corollary, we obtain explicit extractors for APs, GAPs, and Bohr sources with linear min-entropy, although again our results over $\mathbb{Z}_p^n$ require the list-decodability condition. We further explore special cases of additive sources. We improve previous constructions of line sources (affine sources of dimension 1), requiring a field of size linear in $n$, rather than $\Omega(n^2)$ by Gabizon and Raz. This beats the non-explicit bound of $\Theta(n \log n)$ obtained by the probabilistic method. We then generalize this result to APs and GAPs

Algebraic and analytic techniques in coding theory

Error correcting codes are designed to tackle the problem of reliable trans- mission of data through noisy channels. A major challenge in coding theory is to efficiently recover the original message even when many symbols of the received data have been corrupted. This is called the unique decoding problem of error correcting codes. More precisely, if the user wants to send K bits, the code stretches K bits to N bits to tolerate errors in the N bits. Then the goal is to recover the original K bits of the message. Often, the receiver requires only a certain part of the message. In such cases, analyzing the entire received data (word) becomes prohibitive. The challenge is to design a local decoder which queries only few locations of the received word and outputs the part of the message required. This is known as local decoding of an error correcting code. The unique decoding problem faces a certain combinatorial barrier. That is, there is a limit to the number of errors it can tolerate in order to uniquely identify the correct message. This is called the unique decoding radius. A major open problem is to understand what happens if one allows for errors beyond this threshold. The goal is to design an algorithm that can recover the right message, or possibly a list of messages (preferably a small number). This is referred to as list decoding of an error correcting code. At the core of many such codes lies polynomials. Polynomials play a fundamental role in computer science with important applications in algorithm design, complexity theory, pseudo-randomness and machine learning. In this dissertation, we improve our understanding of well known classes of codes and discover various properties of polynomials. As an additional consequence, we obtain results in a suite of problems in effective algebraic geometry, including Hilbert’s nullstellensatz, ideal membership problem and counting rational points in a variety.Computer Science