Self-dual codes over $${\mathbb {F}}_2[u]/\langle u^4 \rangle $$ and Jacobi forms over a totally real subfield of $${\mathbb {Q}}(\zeta _8)$$

Abstract

Let $$K={\mathbb {Q}}(\zeta _8)$$ be the cyclotomic field over $${\mathbb {Q}}$$ of the extension degree 4. We give an integral lattice construction on $${\mathbb {Q}}(\zeta _8)$$ induced from codes over the ring $${\mathcal {R}}= {\mathbb {F}}_2[u]/\langle u^4 \rangle $$ . We define a theta series using these lattices and discuss its relation with the complete weight enumerator of a code over $${\mathcal {R}}$$ . If C is a 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 n and genus g can be seen in terms of the complete joint weight enumerator of codes $$C_j$$ , for $$1\le j\le g$$ over $${\mathcal {R}}$$ .