Nonexistence of two classes of generalized bent functions

Abstract

We obtain new nonexistence results of generalized bent functions from Z n q to Z q (called type [n, q]) in the case that there exist cyclotomic integers in Z[ζ q ] with absolute value q n 2 . This result generalize the previous two scattered nonexistence results [n, q] = [1, 2 × 7] of Pei [13] and [3, 2 × 23 e ] of Jiang-Deng [7] to a generalized class. In the last section, we remark that this method can apply to the GBF from Z n 2 to Z m .

Nonexistence of generalized bent functions and the quadratic norm form equations

“…, g, (4) (Feng [3]) type [n < m/s, 2p l ], where n is odd, p ≡ 7 (mod 8) is a prime, s = ϕ(p l ) ord p l (2) (here ϕ is the Euler phi function, and ord N (a) means the order of a in the multiplicative group (Z/N Z) × ) and m is the smallest odd positive integer s.t. x 2 + py 2 = 2 m+2 has integral solutions, (5) (Feng et al [3,4,5]) various classes with type [n < m, 2p l1 1 p l2 2 ], where n is odd, p 1 , p 2 are two distinct primes satisfying some conditions and m is an upper bound for n, (6) (Jiang and Deng [9]) type [3, 2 × 23 e ], (7) (Li and Deng [11]) type [m, 2p e ] where p ≡ 7 (mod 8) is a prime with ord p e (2) = ϕ(p e )/2 and m is defined the same as in ( 4), (8) (Lv and Li [14]) type [m, 2p r1 1 p r2 2 ] where p 1 ≡ 7 (mod 8) and p 2 ≡ 5 (mod 8) are two primes satisfying some conditions and m is defined the same as in (4) except that p is replaced by p 1 , (9) (Lv and Li [14]) type [1 ≤ n ≤ 3, 2 × 31 e ] and [1 ≤ n ≤ 5, 2 × 151 e ] where e and n are positive integers and n is odd.…”