Abstract
In [15], the second author defined algebraic geometric codes over rings. This definition was motivated by two recent trends in coding theory: the study of algebraic geometric codes over finite fields, and the study of codes over rings. In that paper, many of the basic parameters of these new codes were computed. However, the Lee weight, which is very important for codes over the ring ℤ/4ℤ, was not considered. In [14], this weight measure, as well as the more general Euclidean weight for codes over ℤ/p 1ℤ, is considered for algebraic geometric codes arising from elliptic curves. In this paper, we will focus on the specific case of codes over ℤ/4ℤ and we will show how everything works in an explicit example.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
A.R. Calderbank and G.M. McGuire, Construction of a (64, 237, 12) code via Galois rings, Designs, Codes, and Cryptography, vol. 10, pp. 157–165, 1997.
A.R. Hammons Jr., P.V. Kumar, A.R. Calderbank, N.J.A. Sloane, and P. SOLé, The ℤ4-linearity of Kerdock, Preparata, Goet hals, and related codes, IEEE Trans. Inform. Theory, vol.40, pp. 301–319, 1994.
N.M. Katz, Serre-Tate local moduli, in Algebraic Surfaces, (Orsay, 1976–1978), Lecture Notes Math., vol.868, pp. 138–202, Berlin: Springer-Verlag 1981.
P.V. Kumar, T. Helleseth, and A.R. Calderbank, An upper bound for some exponential sums over Galois rings and applications, IEEE Trans. Inform. Theory, vol.41, pp. 456–468, 1995.
S. Lang, Algebra, Reading, MA: Addison-Wesley 1974.
W-C.W. Li, Character sums over p-adic fields, unpublished manuscript, 1997.
J. Lubin, J-P. Serre, and J. Tate, Elliptic curves and formal groups, in Proc. Woods Hole Summer Institute Alg. Geometry, Woods Hole, MA, 1964.
H.L. Schmid, Zur arithmetik der zyklischen p-Körper, Crelles J. vol. 176, pp. 161–167 1936.
—, Kongmenzzetafunktionen in zyklischen Körpern, Abh. Preuss. Akad. Wiss. Math.-Nat. Kl., vol. 1941, 30pp., 1942.
J. Silverman, The Arithmetic of Elliptic Curves, New York: Springer-Verlag 1986.
P. SOLé, A quaternary cyclic code and a family of quadriphase sequences with low correlation, Lect. Notes Comp. Science, vol.388, pp. 193–201, New York: Springer 1989
H. Stichtenoth, Algebraic Function Fields and Codes, Berlin: Springer-Verlag 1993.
M.A. Tsfasman and S.G. Vlăduts, Algebraic-Geometry Codes, Dodrecht: Kluwer Academic 1991.
J.-F. Voloch and J.L. Walker, Euclidean weights of codes from elliptic curves over rings, submitted for publication.
J.L. Walker, Algebraic geometric codes over rings, J. Pure and Appl. Alg., to appear.
—, Algebraic geometric codes over rings, Ph.D. thesis, University of Illinois, 1996.
—, The Nordstrom-Robinson code is algebraic geometric, IEEE Trans. Inform. Theory, vol.43, pp. 1588–1593, 1997.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1998 Springer Science+Business Media New York
About this chapter
Cite this chapter
Voloch, J.F., Walker, J.L. (1998). Lee Weights of Codes from Elliptic Curves. In: Vardy, A. (eds) Codes, Curves, and Signals. The Springer International Series in Engineering and Computer Science, vol 485. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-5121-8_5
Download citation
DOI: https://doi.org/10.1007/978-1-4615-5121-8_5
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4613-7330-8
Online ISBN: 978-1-4615-5121-8
eBook Packages: Springer Book Archive