Type II codes over \(\mathbb F_2+ u\mathbb F_2\) and applications to Hermitian modular forms (Q1429135)

The authors study Type II codes over the ring \(R=F_{2}+uF_{2}\) \(=\mathbb{Z}\left[ i\right]/2\mathbb{Z}\left[ i\right]\) of 4 elements, where \(u^{2}=0\). They connect the algebra of symmetrized biweight enumerators of Type II codes over the ring \(R\) with the algebra of polynomial invariants by a certain finite group action, and then directly construct symmetric Hermitian modular forms from the polynomial invariants by the finite group, by substituting certain theta series for the indeterminates of the polynomial. As an application, they construct Freitag's generators of weight 4, 8, 12, 12, and 16 of the ring of Hermitian modular forms of degree 2. The remaining generator, a weight 10 symmetric Hermitian modular form (in this case, the weight is not multiple of 4), is obtained as an invariant of a finite group \(G\subset\text{GL}(10,\mathbb{C})\) of order 737280.