Gauss sums for O⁺(2n,q)

Identities arising from Gauss sums for finite classical groups

“…Some expressions of Gauss sums (1.1) for various finite classical groups were obtained by using Bruhat decompositions of those groups with respect to certain maximal parabolic subgroups [7][8][9][10][11][12]14]. The main formula (3.1) allows us to get alternative expressions for those Gauss sums.…”

“…On the other hand, alternative expressions of those sums were gotten by using Bruhat decompositions with respect to certain maximal parabolic subgroups of the finite classical groups (cf. [7][8][9][10][11][12]14]). In this section, equating two different expressions of the Gauss sum for each finite classical group, we will be able to produce identities, some of which are new and some old.…”

“…In a series of papers [7][8][9][10][11][12]14] the second author considered Gauss sums for various finite classical groups, explicitly determined their values, and showed, for example, that they can be used in finding formulas for the number of elements with given trace and determinant in various finite classical groups. The same line of study was extended to the 7-dimensional faithful representation of G 2 (q) and the adjoint representation of GL(n, q) [6,16].…”

Gauss sums over some matrix groups

“…The Gauss sums for classical groups over a finite field have been extensively studied by Kim in several articles [9,[6][7][8]11,10,[12][13][14][15][16][17][18][19]1], and more recently, he also applied the Gauss sum for special linear groups over finite fields to coding theory [20].…”

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

“…Since the sum in (1.3) vanishes for r ¼ 1, the polynomials involving (1.3) do not appear in that case. For r ¼ 1, similar sums for other classical groups over a finite field had been considered ( [7]- [12], [16], [17]). …”