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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.
IDENTITIES ARISING FROM GAUSS SUMS FOR SYMPLECTIC AND ORTHOGONAL GROUPS
채희준,김대산 대한수학회 2010 대한수학회지 Vol.47 No.2
We express Gauss sums for symplectic and orthogonal groups over finite fields as averages of exponential sums over certain maximal tori. Together with our previous results, we obtain some interesting identities involving various classical Gauss and Kloosterman sums.
IDENTITIES ARISING FROM GAUSS SUMS FOR SYMPLECTIC AND ORTHOGONAL GROUPS
Chae, Hi-Joon,Kim, Dae-San Korean Mathematical Society 2010 대한수학회지 Vol.47 No.2
We express Gauss sums for symplectic and orthogonal groups over finite fields as averages of exponential sums over certain maximal tori. Together with our previous results, we obtain some interesting identities involving various classical Gauss and Kloosterman sums.
THE GAUSS SUMS OVER GALOIS RINGS AND ITS ABSOLUTE VALUES
Jang, Young Ho,Jun, Sang Pyo The Kangwon-Kyungki Mathematical Society 2018 한국수학논문집 Vol.26 No.3
Let ${\mathcal{R}}$ denote the Galois ring of characteristic $p^n$, where p is a prime. In this paper, we investigate the elementary properties of Gauss sums over ${\mathcal{R}}$ in accordance with conditions of characters of Galois rings, and we restate results for Gauss sums in [1, 2, 3, 7, 12, 13]. Also, we compute the modulus of the Gauss sums.
THE JACOBI SUMS OVER GALOIS RINGS AND ITS ABSOLUTE VALUES
Jang, Young Ho Korean Mathematical Society 2020 대한수학회지 Vol.57 No.3
The Galois ring R of characteristic p<sup>n</sup> having p<sup>mn</sup> elements is a finite extension of the ring of integers modulo p<sup>n</sup>, where p is a prime number and n, m are positive integers. In this paper, we develop the concepts of Jacobi sums over R and under the assumption that the generating additive character of R is trivial on maximal ideal of R, we obtain the basic relationship between Gauss sums and Jacobi sums, which allows us to determine the absolute value of the Jacobi sums.
The Jacobi sums over Galois rings and its absolute values
장영호 대한수학회 2020 대한수학회지 Vol.57 No.3
The Galois ring $\GR$ of characteristic $p^n$ having $p^{mn}$ elements is a finite extension of the ring of integers modulo $p^n$, where $p$ is a prime number and $n,m$ are positive integers. In this paper, we develop the concepts of Jacobi sums over $\GR$ and under the assumption that the generating additive character of $\GR$ is trivial on maximal ideal of $\GR$, we obtain the basic relationship between Gauss sums and Jacobi sums, which allows us to determine the absolute value of the Jacobi sums.
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).
GAUSS SUMS OVER GALOIS RINGS OF CHARACTERISTIC 4
Oh, Yunchang,Oh, Heung-Joon The Kangwon-Kyungki Mathematical Society 2001 한국수학논문집 Vol.9 No.1
In this paper, we define and study Gauss sums over Galois rings of characteristic 4. In particular, we give the absolute value of Gauss sum over Galois rings of characteristic 4.
ON THE GENERAL QUADRATIC GAUSS SUMS WEIGHTED BY CHARACTER SUMS OVER A SHORT INTERVAL
Zhang, Tianping Korean Mathematical Society 2013 대한수학회보 Vol.50 No.3
By using the analytic methods, the mean value of the general quadratic Gauss sums weighted by the first power mean of character sums over a short interval is investigated. Several sharp asymptotic formulae are obtained, which show that these sums enjoy good distributive properties. Moreover, interesting connections among them are established.
On the general quadratic Gauss sums weighted by character sums over a short interval
Tianping Zhang 대한수학회 2013 대한수학회보 Vol.50 No.3
By using the analytic methods, the mean value of the general quadratic Gauss sums weighted by the first power mean of character sums over a short interval is investigated. Several sharp asymptotic formulae are obtained, which show that these sums enjoy good distributive properties. Moreover, interesting connections among them are established.