Codes over \(\Sigma_{2m}\) and Jacobi forms over the quaternions
If \(i,j,k = ij\) denote the standard quaternions the authors consider the ring \(R=\mathbb{Z}/2m\mathbb{Z}+ i\mathbb{Z}/2m\mathbb{Z}+ j\mathbb{Z}/2m\mathbb{Z}+k\mathbb{Z}/ 2m\mathbb{Z}\), \(m\in \mathbb{N}\). They relate self-dual codes over \(R\) to quaternionic unimodular lattices and to self-dual lattices over \(\mathbb{Z}/2m\mathbb{Z}\). The connection is studied between the complete weight enumerators of codes over \(R\) and Jacobi forms on the half-spaces of quaternions (cf. the reviewer [\textit{A. Krieg}, Modular forms on half-spaces of quaternions (Lecture Notes in Mathematics 1143, Berlin, Springer) (1985; Zbl 0564.10032)]. Higher genus modular forms over the quaternions are also constructed using joint weight enumerators of codes.
scientific article; zbMATH DE number 2123506