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

Abstract

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.

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].…”