Codes over $\mathbf {GF\pmb (4\pmb )}$ and $\mathbf {F}_2 \times \mathbf {F}_2$ and Hermitian lattices over imaginary quadratic fields
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- by Kok Seng Chua
- Proc. Amer. Math. Soc. 133 (2005), 661-670
- DOI: https://doi.org/10.1090/S0002-9939-04-07724-X
- Published electronically: September 20, 2004
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Abstract:
We introduce a family of bi-dimensional theta functions which give uniformly explicit formulae for the theta series of hermitian lattices over imaginary quadratic fields constructed from codes over $\textbf {GF(4)}$ and $\mathbf {F}_2 \times \mathbf {F}_2$, and give an interesting geometric characterization of the theta series that arise in terms of the basic strongly $\ell$ modular lattice $\mathbf {Z}+\sqrt {\ell }\mathbf {Z}$. We identify some of the hermitian lattices constructed and observe an interesting pair of nonisomorphic 3/2 dimensional codes over $\textbf {F}_2 \times \mathbf {F}_2$ that give rise to isomorphic hermitian lattices when constructed at the lowest level 7 but nonisomorphic lattices at higher levels. The results show that the two alphabets $\textbf {GF(4)}$ and $\mathbf {F}_2 \times \mathbf {F}_2$ are complementary and raise the natural question as to whether there are other such complementary alphabets for codes.References
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Bibliographic Information
- Kok Seng Chua
- Affiliation: Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543
- Address at time of publication: Software and Computing Programme, Institute of High Performance Computing, 1 Science Park Road, #01-01, The Capricorn, Singapore Science Park II, Singapore 117528
- Email: matcks@nus.edu.sg, chuaks@ihpc.a-star.edu.sg
- Received by editor(s): March 3, 2003
- Received by editor(s) in revised form: October 29, 2003
- Published electronically: September 20, 2004
- Communicated by: Wen-Ching Winnie Li
- © Copyright 2004
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 133 (2005), 661-670
- MSC (2000): Primary 11H71, 94B75; Secondary 11H31
- DOI: https://doi.org/10.1090/S0002-9939-04-07724-X
- MathSciNet review: 2113912