Abstract
We construct a series of algebraic geometric codes using a class of curves which have many rational points. We obtain codes of lengthq 2 over\(\mathbb{F}\) q , whereq = 2q 20 andq 0 = 2n, such that dimension + minimal distance ≧q 2 + 1 − q 0 (q − 1). The codes are ideals in the group algebra\(\mathbb{F}\) q [S], whereS is a Sylow-2-subgroup of orderq 2 of the Suzuki-group of orderq 2 (q 2 + 1)(q − 1). The curves used for construction have in relation to their genera the maximal number of\(\mathbb{F}\) GF q -rational points. This maximal number is determined by the explicit formulas of Weil and is effectively smaller than the Hasse—Weil bound.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Berman, S. D.: On the theory of group codes. Kibernetika3, 31–39 (1967)
Charpin, P.: Codes idéaux de certains algèbres modulaire. These de 3iéme cycle, Universite de Paris VII (1982)
Charpin, P.: The extended Reed—Solomon codes considered as ideals of a modular algebra. Ann. Discrete Math.17, 171–176 (1983)
Charpin, P.: A new description of some polynomial codes: the primitive generalized Reed-Muller codes, preprint
Chevalley, C.: Introduction to the theory of algebraic functions of one variable. Am. Math. Soc. New York: 1951
Damgård, I., Landrock, P.: Ideals and codes in group algebras. Aarhus Universitet, Preprint Series
Goppa, V. D.: Codes an algebraic curves. Dokl. Akad. NAUK, SSSR,259, 1289–1290 (1981); (Soviet Math. Dokl.24, 170–172 (1981))
Goppa, V. D.: Algebraico-geometric Codes, Izv. Akad. NAUK, SSSR,46 (1982); (Math. U.S.S.R. Izvestiya,21, 75–91 (1983))
Hansen, J. P.: Codes on the Klein Quartic, Ideals and decoding. IEEE Trans. on Inform. Theory33 (1987)
Hansen, J. P.: Group codes on algebraic curves. Mathematica Gottingensis, Heft 9, Feb. 1987
Hasse, H.: Theorie der relativ zyklischen algebraischen Funktionenkörper. J. Reine Angew. Math.172, 37–54 (1934)
Henn, H. W.: Funktionenkörper mit grosser Automorphismengruppe. J. Reine Angew. Math.302, 96–115 (1978)
Landrock, P., Manz, O.: Classical codes as ideals in group algebras. Aarhus Universitet, Preprint Series, 1986/87 No. 18
Serre, J. P.: Sur le nombre des points rationels d'une courbe algébrique sur un corps fini. C. R. Acad. Sci. Paris Sér. I Math,296, 397–402 (1983)
Serre, J. P.: Corps locaux, Paris 1962
Stichtenoth, H.: On automorphisms of geometric Goppa codes. J. Alg. (to appear)
Stichtenoth, H.: A note on Hermitian Codes overGF(q 2), IEEE Trans. on Inform. Theory34, 1345–1348 (1988)
Stichtenoth, H.: Self-dual Goppa Codes, J. Pure, Appl. Algebra55, 199–211 (1988)
Tiersma, H. J.: Remarks on codes from Hermitian curves. IEEE Trans. Inform. Theory33, 605–609 (1987)
Tsfasman, M. A., Vladut, S. G., Zink, Th.: Modular curves, Shimura curves and Goppa codes, better than Varshamov—Gilbert bound. Math. Nachr.109, 13–28 (1982)
Author information
Authors and Affiliations
Additional information
Supported by Deutsche Forschungsgemeinschaft while visiting Essen University
Rights and permissions
About this article
Cite this article
Hansen, J.P., Stichtenoth, H. Group codes on certain algebraic curves with many rational points. AAECC 1, 67–77 (1990). https://doi.org/10.1007/BF01810849
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01810849