Binary self-dual codes and Jacobi forms over a totally real subfield of $${\mathbb {Q}}(\zeta _8)$$

Let $$K={\mathbb {Q}}(\zeta _8)$$ be the complex multiplication field over $${\mathbb {Q}}$$ of extension degree 4. We give an integral lattice construction on $${\mathbb {Q}}(\zeta _8)$$ induced from binary codes. We define a theta series using these lattices and discuss its relation with the complete weight enumerator of a binary code. If C is a binary Type II code of length l, we find that the complete weight enumerator of C gives a Jacobi form of weight l and the index 2l over the maximal totally real subfield $$k={\mathbb {Q}}(\zeta _8+\zeta _8^{-1})$$ of K. Also, we see that Hilbert-Siegel modular form of weight l and genus g can be seen in terms of the complete joint weight enumerator for codes $$C_j$$ , $$1\le j\le g$$ over $${\mathbb {F}}_2$$ .