Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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
DOI: https://doi.org/10.1090/S0002-9939-04-07724-X
Published electronically: September 20, 2004
MathSciNet review: 2113912
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • 1. C. Bachoc, Applications of coding theory to the construction of modular lattices, J. Combinatorial Theory, Ser. A, 78 (1997) 92-119. MR 98a:11084
  • 2. C. Bachoc, G. Nebe, Zonal functions for the unitary groups and applications to hermitian lattices, J. Number Theory, 96 (2002), No-1, 55-75.MR 2003i:33016
  • 3. M. Broué, M. Enguehard, Polyn$\hat{o}$mes des poids de certains codes et fonctions thêta de certains réseaux, Ann. Sc. ENS 5 (1972) 157-181. MR 48:3596
  • 4. H. H. Chan, K. S. Chua, P. Solé, Quadratic Iterations to $\pi$ associated with elliptic functions to the cubic and septic base, Trans. AMS 355 (2003), 1505-1520. MR 2003j:33008
  • 5. H. H. Chan, K. S. Chua, P. Solé, Seven-modular lattices and a septic base Jacobi Identity, Journal of Number Theory, Vol 99 No. 2, (2003), 361-372.MR 2003m:11102
  • 6. K. S. Chua, P. Solé, Jacobi Identities, modular lattices, and modular towers, to appear in European Journal of Combinatorics, 25 No. 4 (2004), 495-503.
  • 7. J. Conway, V. Pless, N. Sloane, Self-Dual Codes over $GF(3)$ and $GF(4)$ of Length not Exceeding 16, IEEE Trans on Info. Theory, Vol IT-25, No 3, May 1979.MR 80h:94026
  • 8. J. Conway, N. Sloane, Sphere Packings, Lattices and Groups, 3rd Edition, Springer 1991.MR 93h:11069
  • 9. W. Feit, Some Lattices over $Q(\sqrt{-3})$, Journal of Algebra, 52 (1978),248-263.MR 58:16534
  • 10. Nebe-Sloane Catalogue of lattices, http://www.research.att.com/ njas/lattices/ P16.2.html
  • 11. H.-G. Quebbemann, Modular Lattices in Euclidean Spaces, Journal of Number Theory 54 (1995) , 190-202.MR 96i:11072
  • 12. H.- G. Quebbemann, Atkin-Lehner eigenforms and strongly modular lattices, L'Enseignement Mathématique, t. 43 (1997), 55-65.MR 98g:11081
  • 13. E. Rains, N. Sloane, The Shadow Theory of Modular and Unimodular Lattices, Journal of Number Theory, Vol 73 (1998), 359-389.MR 99i:11053
  • 14. Eric M. Rains and N. J. A. Sloane, Self-dual codes, Handbook of coding theory, Vol. I, II, North-Holland, Amsterdam, 1998, pp. 177–294. MR 1667939
  • 15. R. Schulze-Pillot, Genera of hermitian lattices over imaginary quadratic fields http://www.math. uni-sb.de/ag-schulze/Hermitian-lattices/
  • 16. R. Scharlau, R. Schulze-Pillot, Extremal Lattices, Algorithmic algebra and number theory, Heidelberg, Springer, Berlin, (1999), 139-170. MR 99m:11074
  • 17. R. Scharlau, A. Schiemann, R. Schulze-Pillot, Theta Series of Modular, Extremal, and Hermitian Lattices, Integral quadratic forms and lattices (Seoul, 1998), 221-233, Contemp. Math., 249, Amer. Math. Soc., Providence, RI, 1999. MR 2001a:11066
  • 18. N. J. A. Sloane, Codes over $GF(4)$ and Complex Lattices, Journal of Algebra, 52 (1978), 168-181.MR 58:9782

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 11H71, 94B75, 11H31

Retrieve articles in all journals with MSC (2000): 11H71, 94B75, 11H31


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
Email: matcks@nus.edu.sg, chuaks@ihpc.a-star.edu.sg

DOI: https://doi.org/10.1090/S0002-9939-04-07724-X
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.

American Mathematical Society