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 \({\mathbb {F}}_2\).

令\（K = {\ mathbb {Q}}（\ zeta _8）\）是扩展度为4的\（{\ mathbb {Q}} \\）的复数乘法字段。我们在\（由二进制代码产生的{\ mathbb {Q}}（\ zeta _8）\）。我们使用这些晶格定义theta级数，并与二进制代码的完整权重枚举数讨论其关系。如果C是长度为l的二进制II型代码，则我们发现C的完整权重枚举器在最大的完全实子字段\（k = {\ mathbb {Q}}上给出了权重l的Jacobi形式和索引2 l。（\ zeta _8 + \ zeta _8 ^ {-1}）\）的K。同样，我们看到权重l和属g的希尔伯特-西格尔模块化形式可以从\（C_j \），\（1 \ le j \ le g \）上\（{ \ mathbb {F}} _ 2 \）。