Exponential sums for O<sup>-</sup>(2n,q) and their applications

Ким, Дае Сан

doi:10.4064/aa97-1-4

Cited by 7 publications

(3 citation statements)

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“…First, we collect some results from [14] with one modification. For a nontrivial additive character of F q , a nonnegative integer m, and with ; r as above, we have ðm; q; rÞ is as in (4.27) and (4.28), and the sum in (5.5) is over all y 2 F q with y r ¼ .…”

Section: A Propositionmentioning

confidence: 99%

“…We will show that (1.1) is a polynomial in q times X 2F q ð r Þ ð 1:3Þ plus another polynomial in q involving certain exponential sums (cf. (2.14) (2.15)), of which O-estimates were given in [14]. On the other hand, the expression for (1.2) is that for (1.1) plus ðÀ1Þ times a similar one corresponding to the subsum of (1.2) over Oð2n þ 1; qÞ À SOð2n þ 1; qÞ ¼ SOð2n þ 1; qÞ (cf.…”

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confidence: 99%

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Exponential sums for O(2n+1,q) and their applications

Ким¹

2001

Glasgow Math. J.

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For a nontrivial additive character and a multiplicative character of the finite field with q elements (q a power of an odd prime), and for each positive integer r, the exponential sums P ððtr wÞ r Þ over w 2 SOð2n þ 1; qÞ and P ðdet wÞððtr wÞ r Þ over Oð2n þ 1; qÞ are considered. We show that both of them can be expressed as polynomials in q involving certain exponential sums. Also, from these expressions we derive the formulas for the number of elements w in SOð2n þ 1; qÞ and Oð2n þ 1; qÞ with ðtr wÞ r ¼ , for each in the finite field with q elements.

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Section: A Propositionmentioning

confidence: 99%

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confidence: 99%

Exponential sums for O(2n+1,q) and their applications

Ким¹

2001

Glasgow Math. J.

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“…The Gauss sums for classical groups over a finite field have been extensively studied by Kim in several articles [8,5,6,7,9,10,11,12,13,14,15,16,17,18], and more recently, he also applied the Gauss sum for special linear groups over finite fields to coding theory [19].…”

Section: Introductionmentioning

confidence: 99%

Gauss sums over some matrix groups

Li¹,

2012

Journal of Number Theory

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In this note, we give explicit expressions of Gauss sums for general (resp. special) linear groups over finite fields, which involve classical Gauss sums (resp. Kloosterman sums). The key ingredient is averaging such sums over Borel subgroups, i.e., the groups of upper triangular matrices. As applications, we count the number of invertible matrices of zero-trace over finite fields and we also improve two bounds of Ferguson, Hoffman, Luca, Ostafe and Shparlinski in [R. Ferguson, C. Hoffman, F. Luca, A. Ostafe, I.E. Shparlinski, Some additive combinatorics problems in matrix rings, Rev. Mat. Complut. 23 (2010) 501-513].

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