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ON THE HYBRID MEAN VALUE OF GENERALIZED DEDEKIND SUMS, GENERALIZED HARDY SUMS AND KLOOSTERMAN SUMS
Qing Tian,Yan Wang Korean Mathematical Society 2023 대한수학회보 Vol.60 No.3
The main purpose of this paper is to study the hybrid mean value problem involving generalized Dedekind sums, generalized Hardy sums and Kloosterman sums. Some exact computational formulas are given by using the properties of Gauss sums and the mean value theorem of the Dirichlet L-function. A result of W. Peng and T. P. Zhang [12] is extended. The new results avoid the restriction that q is a prime.
김대산 장전수학회 2015 Advanced Studies in Contemporary Mathematics Vol.25 No.2
In this paper, we construct two innite families of binary linear codes associated with double cosets with respect to certain maximal parabolic subgroup of the symplectic group Sp(2n; q). Here q is a power of two. Then we obtain an innite family of recursive formulas for the power moments of Kloosterman sums and those of 2-dimensional Kloosterman sums in terms of the frequencies of weights in the codes. This is done via Pless power moment identity and by utilizing the explicit expressions of exponential sums over those double cosets related to the evaluations of \Gauss sums" for the symplectic groups Sp(2n; q).
김대산 대한수학회 2020 대한수학회지 Vol.57 No.3
In this paper, we construct four infinite families of binary linear codes associated with double cosets with respect to a certain maximal parabolic subgroup of the orthogonal group $O^{+}(2n,2^{r})$. And we obtain two infinite families of recursive formulas for the power moments of Kloosterman sums and those of 2-dimensional Kloosterman sums in terms of the frequencies of weights in the codes. This is done via Pless' power moment identity and by utilizing the explicit expressions of exponential sums over those double cosets related to the evaluations of ``Gauss sums" for the orthogonal groups $O^{+}(2n,2^{r})$.
Kim, Dae San Korean Mathematical Society 2020 대한수학회지 Vol.57 No.3
In this paper, we construct four infinite families of binary linear codes associated with double cosets with respect to a certain maximal parabolic subgroup of the orthogonal group O<sup>+</sup>(2n, 2<sup>r</sup>). And we obtain two infinite families of recursive formulas for the power moments of Kloosterman sums and those of 2-dimensional Kloosterman sums in terms of the frequencies of weights in the codes. This is done via Pless' power moment identity and by utilizing the explicit expressions of exponential sums over those double cosets related to the evaluations of "Gauss sums" for the orthogonal groups O<sup>+</sup>(2n, 2<sup>r</sup>).
GAUSS SUMS FOR U(2n + 1,$q^2$)
Kim, Dae-San Korean Mathematical Society 1997 대한수학회지 Vol.34 No.4
For a lifted nontrivial additive character $\lambda'$ and a multiplicative character $\chi$ of the finite field with $q^2$ elements, the 'Gauss' sums $\Sigma\lambda'$(tr $\omega$) over $\omega$ $\in$ SU(2n + 1, $q^2$) and $\Sigma\chi$(det $\omega$)$\lambda'$(tr $\omega$) over $\omega$ $\in$ U(2n + 1, $q^2$) are considered. We show that the first sum is a polynomial in q with coefficients involving certain new exponential sums and that the second one is a polynomial in q with coefficients involving powers of the usual twisted Kloosterman sums and the average (over all multiplicative characters of order dividing q-1) of the usual Gauss sums. As a consequence we can determine certain 'generalized Kloosterman sum over nonsingular Hermitian matrices' which were previously determined by J. H. Hodges only in the case that one of the two arguments is zero.
Kim, Dae-San Korean Mathematical Society 2011 대한수학회지 Vol.48 No.2
In this paper, we construct eight infinite families of ternary linear codes associated with double cosets with respect to certain maximal parabolic subgroup of the special orthogonal group $SO^-$(2n, q). Here q is a power of three. Then we obtain four infinite families of recursive formulas for power moments of Kloosterman sums with square arguments and four infinite families of recursive formulas for even power moments of those in terms of the frequencies of weights in the codes. This is done via Pless power moment identity and by utilizing the explicit expressions of exponential sums over those double cosets related to the evaluations of "Gauss sums" for the orthogonal groups $O^-$(2n, q).
김대산 대한수학회 2011 대한수학회지 Vol.48 No.2
In this paper, we construct eight innite families of ternary linear codes associated with double cosets with respect to certain maximal parabolic subgroup of the special orthogonal group SO^-(2n, q). Here q is a power of three. Then we obtain four innite families of recursive formulas for power moments of Kloosterman sums with square arguments and four innite families of recursive formulas for even power moments of those in terms of the frequencies of weights in the codes. This is done via Pless power moment identity and by utilizing the explicit expressions of exponential sums over those double cosets related to the evaluations of "Gauss sums" for the orthogonal groups O^-(2n, q).
김대산,J. H. Kim 장전수학회 2012 Proceedings of the Jangjeon mathematical society Vol.15 No.1
In this paper, we construct two ternary linear codes associated with the symplectic groups Sp(2, q) and Sp(4, q). Here q is a power of three. Then we obtain recursive formulas for the power moments of Kloosterman sums with square arguments and for the even power moments of those in terms of the frequencies of weights in the codes. This is done via Pless power moment identity and by utilizing the explicit expressions of “Gauss sums” for the symplectic groups Sp(2n, q).
ON THE r-TH HYPER-KLOOSTERMAN SUMS AND ITS HYBRID MEAN VALUE
Zhang, Tianping,Zhang, Wenpeng Korean Mathematical Society 2006 대한수학회지 Vol.43 No.6
The main purpose of this paper is using the properties of Gauss sums, primitive characters and the mean value theorems of Dirichlet L-functions to study the hybrid mean value of the T-th hyper-Kloosterman sums Kl(h, k+1, r;q) and the hyper Cochrane sums C(h, q; m, k), and give an interesting mean value formula.
HYBRID MEAN VALUE OF GENERALIZED BERNOULLI NUMBERS, GENERAL KLOOSTERMAN SUMS AND GAUSS SUMS
Liu, Huaning,Zhang, Wenpeng Korean Mathematical Society 2007 대한수학회지 Vol.44 No.1
The main purpose of this paper is to use the properties of primitive characters, Gauss sums and Ramanujan's sum to study the hybrid mean value of generalized Bernoulli numbers, general Kloosterman sums and Gauss sums, and give two asymptotic formulae.