\(U_n\) lattices, construction \(B\), and AGM iterations (Q1268383)

Following \textit{J. M. Borwein} and \textit{P. B. Borwein} [Trans. Am. Math. Soc. 323, 691-701 (1991; Zbl 0725.33014)] the authors consider generalized arithmetic-geometric mean (AGM) iterations. From the paper: ``We suggest an approach based on codes of length \(L\) over an alphabet of size \(s\) to parametrize the general AGM of order \(L\) and parameter \(s\). 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''.