A characterization of finite commutative Frobenius rings and applications to algebraic coding theory

The fundamental alphabets for codes over rings are finite Frobenius rings, largely because their generating character produces MacWilliams relations. We prove that the following are equivalent statements for a finite commutative ring $ R $: (1) $ R $ is Frobenius; (2) $ | {\mathfrak{a}}| | {\mathfrak{a}}^\perp| = |R| $ for all ideals $ {\mathfrak{a}} $ in $ R $; and (3) $ ( {\mathfrak{a}}^\perp)^\perp = {\mathfrak{a}} $ for all ideals $ {\mathfrak{a}} $ in $ R $ and give an algorithmic way of producing the generating character.