Quadratic Gauss Sums

Abstract

Let p be a prime integer and m be an integer, not divisible by p. Let K be the splitting field of XK!1 over the prime field % N . Solving the Gauss sums problem of order m in characteristic p means determining Gauss sums of all multiplicative characters of K of order dividing m. Our aim is to solve this problem when the subgroup 1p2 is of index 2 in (9/m9)*.1998 Academic Press

All Two-Weight Irreducible Cyclic Codes?

“…To prove the following theorem, we combine Louboutin's bound with work of Baumert and Mykkelveit [4] and recent work of Mbodj [14] on Gauss sums. Otherwise, we may use Proposition 3.8 of [14] applied to a character of GF(pE) of order rs in the estimation of (u, p).…”

“…Otherwise, we may use Proposition 3.8 of [14] applied to a character of GF(pE) of order rs in the estimation of (u, p). Here g"(r!1)(s!1)/2.…”

“…Following Mbodj [14], we say that the pair (u, p) falls under the index 2 case if u is odd and ord S (p)" (u)/2. Note that u can have at most two distinct prime divisors in this case.…”

Complete solving of explicit evaluation of Gauss sums in the index 2 case

“…It is called the "index r case". In 1970's-2000's, the Gauss sums in the index r =2 case have been studied and evaluated explicitly in a series of papers [9,[12][13][14]17]. And since 2005, the case of index r = 4 has been studied [5,6,19].…”

“…In 1997, Langevin [9], as the generalization of [13], gave the evaluation of Gauss sums in the index 2 case for N = l r , where l is an odd prime, r 1. One year later, Mbodj [12], as the generalization of [17], gave the evaluation of Gauss sums in the index 2 case for N = l r1 1 l r2 2 , where l 1 , l 2 are distinct odd primes, r 1 , r 2 1. For N being power of 2, i.e., N = 2 t (t 3), Meijer and van der Vlugt [14], in 2003, evaluated the Gauss sums in the index 2 case and applied the results of Gauss sums to solving the problem of calculating the number of rational points for some algebraic curves over finite field.…”