Codes over Sigma(2m) and Jacobi forms over the Quaternions

Abstract

We introduce codes over the ring Z(2m) + alphaZ(2m) + betaZ(2m) + gammaZ(2m). We relate self-dual codes over this ring to quaternionic unimodular lattices and to self-dual codes over Z(2m) via a gray map. We study a connection between the complete weight enumerators of codes over the quaternionic ring Sigma(2m) and Jacobi forms over the half-space of quaternions. This motivates us to construct an algebra homomorphism from a certain invariant polynomial ring, where the complete weight enumerators belong, to the ring of Jacobi forms over the quaternions. Higher genus modular forms over the quaternions are also constructed using joint weight enumerators of codes.