Several Generalizations of Weil Sums

Chung, Fan

doi:10.1006/jnth.1994.1084

Cited by 15 publications

(10 citation statements)

References 33 publications

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“…Let A, B be arbitrary subsets of the field F p , and χ a non-principal Dirichlet character modulo p. Many authors have studied the following double character sum (see for example [12,14,18,30])…”

Section: Theorem 14 ([40]mentioning

confidence: 99%

Gauss sums and the maximum cliques in generalized Paley graphs of square order

Yip

2021

Preprint

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Let GP (q, d) be the d-Paley graph defined on the finite field F q . It is notoriously difficult to improve the trivial upper bound √ q on the clique number of GP (q, d). In this paper, we investigate the connection between Gauss sums over a finite field and maximum cliques of their corresponding generalized Paley graphs. In particular, we show that the trivial upper bound on the clique number of GP (q, d) attains if and only d | ( √ q + 1), which strengthens the previous related results by Broere-Döman-Ridley and Schneider-Silva, as well as improves the trivial upper bound on the clique number of GP (q, d) when d ∤ ( √ q + 1).

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Section: Theorem 14 ([40]mentioning

confidence: 99%

Gauss sums and the maximum cliques in generalized Paley graphs of square order

Yip

2021

Preprint

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“…As we noted earlier, JL constructions of RIP matrices necessarily use at least Ω δ (M ) = Ω δ (K log(N/K)) random bits, whereas we believe our Legendre symbol construction can be derandomized quite a bit more: At its heart, Conjecture 3.3 is a statement about how well the Legendre symbol exhibits additive cancellations. In particular, if one is willing to use flat restricted orthogonality as a proof technique for RIP, the statement is essentially a bound on incomplete sums of Legendre symbols, much like those investigated in [16]. We had difficulty proving these cancellations, and so we injected some randomness to enable the application of Theorem 3.1.…”

Section: Our Approachmentioning

confidence: 99%

Derandomizing Restricted Isometries via the Legendre Symbol

et al. 2015

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Abstract. The restricted isometry property (RIP) is an important matrix condition in compressed sensing, but the best matrix constructions to date use randomness. This paper leverages pseudorandom properties of the Legendre symbol to reduce the number of random bits in an RIP matrix with Bernoulli entries. In this regard, the Legendre symbol is not special-our main result naturally generalizes to any small-bias sample space. We also conjecture that no random bits are necessary for our Legendre symbol-based construction.

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“…Chung [11] has given some generalisations of the character sum estimates by Weil and Burgess, with applications to the discrepancy of finite graphs, including the Paley graphs; for any graph this is the maximum, over all s, of the difference between the maximum number of edges of an s-vertex subgraph and the average for that s. Estimating character sums is a major activity; Paley himself was an early contributor in [22], but this was in connection with number theory (specifically Dirichlet series), not graph theory.…”

mentioning

confidence: 99%