Non-trivial,self-dual abelian codes of length n where n is odd over Z_4 exist if and only if n is either divisible by a prime r≡-1(mod 8),or a prime r≡1(mod 8) where ord_r (2)is odd,or a prime p and a prime q where ord_p(2)=2~li,ord_q(2)=2~lj,l≥ 1,i is odd,j is even.

Let G be a finite Abelian group,an ideal in the groupring Z_(p~r)[G] is called an Abelian code over Z_(p~r), where Z_(p~r) is the ring of integers modulo p~r.

Rushanan(1991) generalized them to duadic abelian codes and Zhu(1996) further generalized them to duadic group algebra codes.

Let G be a finite Abelian group,an ideal in the grouping Zpr is called an Abelian code over Zpr.

The security~ of these cryptosystems is based on the dificulty in solving descrete logarithms with Abelian group.

The ellipse rotating symmetric group is proposed,which is an Abelian group.

Infrared Abelization of Yang-Mills Theory via Abelian Higgs Variables;

To Demonstate the Essentiol Condition of Abel s Methad of Apprisal in the Infinitesimal Caleulus of Fiyure Item;

This dissertation regards analytic approach of the primitive literature as the research approach and relies mainly on studying Abel\'s French thesis carefully and investigates the course that the oval total mark transforms into oval function in the 18-19th century emphatically.