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 \({\mathcal {R}}\).
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Communicated by J. Bierbrauer.
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The work is supported by the Department of Science and Technology-Science and Energy Research Board, India (DST-SERB, Grant No. YSS/2015/001801)
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Ankur, Kewat, P.K. Self-dual codes over \({\mathbb {F}}_2[u]/\langle u^4 \rangle \) and Jacobi forms over a totally real subfield of \({\mathbb {Q}}(\zeta _8)\). Des. Codes Cryptogr. 89, 1091–1109 (2021). https://doi.org/10.1007/s10623-021-00860-0
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DOI: https://doi.org/10.1007/s10623-021-00860-0