Deligne-Lusztig varieties and group codes

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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.