Abstract
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 \({\mathbb {F}}_2\).
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Ankur, Kewat, P.K. Binary self-dual codes and Jacobi forms over a totally real subfield of \({\mathbb {Q}}(\zeta _8)\). AAECC 34, 377–392 (2023). https://doi.org/10.1007/s00200-021-00509-4
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DOI: https://doi.org/10.1007/s00200-021-00509-4