Abstract
We construct algebraic geometric codes using the Deligne-Lusztig varieties [De-Lu] associated to a connected reductive algebraic group G defined over a finite field \(\mathbb{F}_q \), with Frobenius map F. The codes are obtained as geometric Goppa codes, that is linear error-correcting codes constructed from algebraic varieties [Go1] and [Go2]. The finite group G F of Lie type acts as \(\mathbb{F}_q \)-rational automorphisms on the codes and they become modules over the group algebra \(\mathbb{F}_q \)[G F ]. Algebraic geometric codes with a group algebra structure induced from automorphisms of the underlying variety have been constructed and studied in [Ha1], [Ha2], [Ha-St] and [V].
The Deligne-Lusztig varieties used in the construction of the codes have in some cases many \(\mathbb{F}_q \)-rational points, which ensures that the codes have a large word length. In case G is of type 2 A 2 the Deligne-Lusztig curve considered have 1+q 3 points over \(\mathbb{F}_{q^2 } \). In case G is a Suzuki group 2 B 2, respectively a Ree group 2 G 2, the Deligne-Lusztig curves considered have 1+q 2, respectively 1 + q 3, points over \(\mathbb{F}_q \). In relation to their genera these numbers are maximal as determined by the “explicit formulas” of Weil.
Supported by the Danish National Science Foundation.
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Hansen, J.P. (1992). Deligne-Lusztig varieties and group codes. In: Stichtenoth, H., Tsfasman, M.A. (eds) Coding Theory and Algebraic Geometry. Lecture Notes in Mathematics, vol 1518. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0087993
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DOI: https://doi.org/10.1007/BFb0087993
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