Generalized bent functions and class group of imaginary quadratic fields

Nonexistence of generalized bent functions from $$\mathbb {Z}_{2}^{n}$$ Z 2 n to $$\mathbb {Z}_{m}$$ Z m

“…Let L be a number field. Any nonzero ideal A of O L can be uniquely (up to the order) expressed as A = P a 1 1 · · · P a s s , …”

“…At the beginning of the paper [1], the author presented an explicit description on odd integers m ≥ 3 satisfying the semiprimitive condition (5) (for m = 1, this condition is true automatically). For each nonzero integer d, let d = 2 t d where t ≥ 0 and 2 d .…”

“…Therefore, we get the following result. 1 1 · · · p a s s where p i are distinct odd prime numbers and a i ≥ 1. Then there is no (m, n)-GBF and (2m, n)-GBF for all odd integers n ≥ 1 provided one of the following conditions is satisfied.…”

“…(2) m = p a 1 1 · · · p a s s , 3 ≤ p 1 < p 2 < · · · < p s and 2 n < p 1 + p 2 ; (3) m = p a 1 1 p a 2 2 , p 1 = p 2 , a 1 , a 2 ≥ 1, 2 n = p 1 p 2 − m 1 p 1 − m 2 p 2 for some positive integers m 1 and m 2 .…”

“…Let m = p a 1 1 · · · p a s s (s ≥ 1) where p i are distinct odd prime numbers. Then there exists l ≥ 1 such that 2 l ≡ −1 (mod m) if and only if V 2 (d i ) ≥ 1 is independent of i.…”