تقرير
On the Paley RIP and Paley graph extractor
| العنوان: | On the Paley RIP and Paley graph extractor |
|---|---|
| المؤلفون: | Satake, Shohei |
| سنة النشر: | 2024 |
| المجموعة: | Computer Science Mathematics |
| مصطلحات موضوعية: | Mathematics - Combinatorics, Computer Science - Discrete Mathematics, Computer Science - Information Theory, Mathematics - Number Theory, 05D10, 68R05, 11T24 |
| الوصف: | Constructing explicit RIP matrices is an open problem in compressed sensing theory. In particular, it is quite challenging to construct explicit RIP matrices that break the square-root bottleneck. On the other hand, providing explicit $2$-source extractors is a fundamental problem in theoretical computer science, cryptography and combinatorics. Nowadays, there are only a few known constructions for explicit $2$-source extractors (with negligible errors) that break the half barrier for min-entropy. In this paper, we establish a new connection between RIP matrices breaking the square-root bottleneck and $2$-source extractors breaking the half barrier for min-entropy. Here we focus on an RIP matrix (called the Paley ETF) and a $2$-source extractor (called the Paley graph extractor), where both are defined from quadratic residues over the finite field of odd prime order $p\equiv 1 \pmod{4}$. As a main result, we prove that if the Paley ETF breaks the square-root bottleneck, then the Paley graph extractor breaks the half barrier for min-entropy as well. Since it is widely believed that the Paley ETF breaks the square-root bottleneck, our result accordingly provides a new affirmative intuition on the conjecture for the Paley graph extractor by Benny Chor and Oded Goldreich. Comment: 10 pages, references are updated, comments are welcome |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2405.08608 |
| رقم الأكسشن: | edsarx.2405.08608 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |