2 research outputs found
Bias vs structure of polynomials in large fields, and applications in effective algebraic geometry and coding theory
Let be a polynomial of degree in variables over a finite field
. The polynomial is said to be unbiased if the distribution of
for a uniform input is close to the uniform
distribution over , and is called biased otherwise. The polynomial
is said to have low rank if it can be expressed as a composition of a few lower
degree polynomials. Green and Tao [Contrib. Discrete Math 2009] and Kaufman and
Lovett [FOCS 2008] showed that bias implies low rank for fixed degree
polynomials over fixed prime fields. This lies at the heart of many tools in
higher order Fourier analysis. In this work, we extend this result to all prime
fields (of size possibly growing with ). We also provide a generalization to
nonprime fields in the large characteristic case. However, we state all our
applications in the prime field setting for the sake of simplicity of
presentation.
As an immediate application, we obtain improved bounds for a suite of
problems in effective algebraic geometry, including Hilbert nullstellensatz,
radical membership and counting rational points in low degree varieties.
Using the above generalization to large fields as a starting point, we are
also able to settle the list decoding radius of fixed degree Reed-Muller codes
over growing fields. The case of fixed size fields was solved by Bhowmick and
Lovett [STOC 2015], which resolved a conjecture of Gopalan-Klivans-Zuckerman
[STOC 2008]. Here, we show that the list decoding radius is equal the minimum
distance of the code for all fixed degrees, even when the field size is
possibly growing with
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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