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Published in: Designs, Codes and Cryptography 5/2021

27-03-2021

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

Authors: Ankur, Pramod Kumar Kewat

Published in: Designs, Codes and Cryptography | Issue 5/2021

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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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Metadata
Title
Self-dual codes over and Jacobi forms over a totally real subfield of
Authors
Ankur
Pramod Kumar Kewat
Publication date
27-03-2021
Publisher
Springer US
Published in
Designs, Codes and Cryptography / Issue 5/2021
Print ISSN: 0925-1022
Electronic ISSN: 1573-7586
DOI
https://doi.org/10.1007/s10623-021-00860-0

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