UnLattices, ConstructionB, and AGM Iterations

The quadratic and cubic arithmetic geometric means (AGMs) are known to be parametrized by Jacobian and bidimensional theta series respectively. We suggest an approach based on codes of lengthLover an alphabet of sizesto parametrize the general AGM of orderLand parameters. This is possible given the existence of number fields and quaternion algebras equipped with suitable quadratic forms. Expressions for theta series of lattices over the Gauss, Eisenstein and Hurwitz integers using Hamming weight enumerators of, respectively, binary, ternary and quaternary codes are derived.