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Codes over $\mathbf{GF\pmb(4\pmb)}$ and $\mathbf{F}_2 \times \mathbf{F}_2$ and Hermitian lattices over imaginary quadratic fields

Author: Kok Seng Chua
Journal: Proc. Amer. Math. Soc. 133 (2005), 661-670
MSC (2000): Primary 11H71, 94B75; Secondary 11H31
Published electronically: September 20, 2004
MathSciNet review: 2113912
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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 ${\bf GF(4)}$ 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 ${\bf F}_2 \times \mathbf{F}_2$ 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 ${\bf GF(4)}$ 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.

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Additional 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

Keywords: Codes over rings, Hermitian lattices, theta series
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
Article copyright: © Copyright 2004 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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