A generalization of the Giulietti–Korchmáros maximal curve

Abstract

We introduce a family of algebraic curves over F q 2n (for an odd n) and show that they are maximal. When n = 3, our curve coincides with the F q 6-maximal curve that has been found by Giulietti and Korchmáros recently. Their curve (i.e., the case n = 3) is the first example of a maximal curve proven not to be covered by the Hermitian curve.

Two-Point Codes for the Generalized GK Curve

“…More precisely, a maximal curve of genus g defined over a finite field F q with q elements, has q + 1 + 2 √ qg F qrational points, i.e., points defined over F q ; this only makes sense if the cardinality q is a square number. An important example of a maximal curve is the Hermitian curve, but recently other maximal curves have been described [11,9], often called the generalized Giulietti-Korchmáros (GK) curves. In this article we continue the study of two-point AG codes coming from the generalized GK curves that was initiated in [5].…”

AG codes from the second generalization of the GK maximal curve

“…The automorphisms group of H q is very large compared to g(H q ). Indeed it is isomorphic to PGU(3, q) and its order is larger than 16g(H q ) 4 . Moreover H q has the largest genus admissible for an F q 2 -maximal curve and it is the unique curve having this property up to birational isomorphism, see [15].…”

“…For any odd n ≥ 3 the curve GK 2,n is F q 2n -maximal and for n = 3 is isomorphic to the well-known GK maximal curve constructed by Giulietti and Korchmáros in [5]. The GK curve was already generalized by Garcia, Güneri and Stichtenoth [4] to an infinite family of F q 2n -maximal curves GK 1,n where n ≥ 3 is an odd prime. For this reason we will refer to the maximal curve GK 2,n as the second generalization of the GK maximal curve.…”

AG codes and AG quantum codes from cyclic extensions of the Suzuki and Ree curves

“…and for this reason they have been used in a number of works. Examples of such curves are the Hermitian curve, the GK curve [14], the GGS curve [12], the Suzuki curve [8], the Klein quartic when √ q ≡ 6 (mod 7) [37], together with their quotient curves. Maximal curves often have large automorphism groups which in many cases can be inherited by the code: this can bring good performances in encoding [27] and decoding [19].…”