All reports by Author Silas Richelson:

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TR23-088
| 1st June 2023
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Parker Newton, Silas Richelson, Chase Wilson#### A High Dimensional Goldreich-Levin Theorem

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TR22-069
| 28th April 2022
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Silas Richelson, Sourya Roy#### List-Decoding Random Walk XOR Codes Near the Johnson Bound

Revisions: 1

Parker Newton, Silas Richelson, Chase Wilson

In this work we prove a high dimensional analogue of the beloved Goldreich-Levin theorem (STOC 1989). We consider the following algorithmic problem: given oracle access to a function $f:\mathbb{Z}_q^m\rightarrow\mathbb{Z}_q^n$ such that ${\rm Prob}_{{\bf x}\sim\mathbb{Z}_q^m}\bigl[f({\bf x})={\bf Ax}\bigr]\geq\varepsilon$ for some ${\bf A}\in\mathbb{Z}_q^{n\times m}$ and $\varepsilon>0$, recover ${\bf A}$ (or a list of ... more >>>

Silas Richelson, Sourya Roy

In a breakthrough result, Ta-Shma described an explicit construction of an almost optimal binary code (STOC 2017). Ta-Shma's code has distance $\frac{1-\varepsilon}{2}$ and rate $\Omega\bigl(\varepsilon^{2+o(1)}\bigr)$ and thus it almost achieves the Gilbert-Varshamov bound, except for the $o(1)$ term in the exponent. The prior best list-decoding algorithm for (a variant of) ... more >>>