The automorphism groups of a family of maximal curves

Guralnick, Robert M.; Malmskog, Beth; Pries, Rachel

doi:10.1016/j.jalgebra.2012.03.036

Cited by 16 publications

(21 citation statements)

References 18 publications

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“…As mentioned previously C 1 = H, the Hermitian function field, and C 3 = C, the GK function field. The automorphism group B := Aut(C n ) of C n has been determined in [12,13]. The stabilizer of Q ∞ , which we will denote by B(Q ∞ ), is in most cases equal to the entire automorphism group.…”

Section: Results About the Hermitian Function Fieldmentioning

confidence: 99%

A complete characterization of Galois subfields of the generalized Giulietti–Korchmáros function field

Anbar¹,

Bassa

Beelen³

2017

Finite Fields and Their Applications

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Section: Results About the Hermitian Function Fieldmentioning

confidence: 99%

A complete characterization of Galois subfields of the generalized Giulietti–Korchmáros function field

Anbar¹,

Bassa

Beelen³

2017

Finite Fields and Their Applications

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“…In this section a more detailed description of the results obtained in the previous sections is given for the particular case q = 2, n = 5. Recall that in this case H(P ∞ ) = {0, 8,16,22,24,30,32,33,38,40,41,44,46,48,49 For the point P 0 (and hence for any F q 2 -rational point), we have from Proposition 4.3 H(P 0 ) = {0, 21, 22}∪{29, . .…”

Section: Ag Codes On the Ggs Curve For Q = 2 And N =mentioning

confidence: 99%

“…Proof. From [15,16], Aut(GGS(q, n)) = Q ⋊ Σ, where Q = {Q a,b | a, b ∈ F q 2 , a q + a = b q+1 } and Σ = g ζ , with…”

Section: Quantum Codes From One-point Ag Codes On the Ggs Curvesmentioning

confidence: 99%

AG codes and AG quantum codes from the GGS curve

Bartoli

Montanucci

Zini

2017

Des. Codes Cryptogr.

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In this paper, algebraic-geometric (AG) codes associated with the GGS maximal curve are investigated. The Weierstrass semigroup at all F q 2 -rational points of the curve is determined; the Feng-Rao designed minimum distance is computed for infinite families of such codes, as well as the automorphism group. As a result, some linear codes with better relative parameters with respect to one-point Hermitian codes are discovered. Classes of quantum and convolutional codes are provided relying on the constructed AG codes. support of G. Maximal curves over F q attain the Hasse-Weil upper bound for the number of F q -rational points with respect to their genus and for this reason they have been used in a number of works. Examples of such curves are the Hermitian curve, the GK curve [12], the GGS curve [10], the Suzuki curve [7], the Klein quartic when √ q ≡ 6 (mod 7) [33], 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 [25] and decoding [17]. Good bounds on the parameters of one-point codes, that is AG codes arising from divisors G of type nP for a point P of the curve, have been obtained by investigating the Weierstrass semigroup at P . These results have been later generalized to codes and semigroups at two or more points; see e.g. [4,5,20,21,27,30,31].AG codes from the Hermitian curve have been widely investigated; see [8,[22][23][24]37,39,40] and the references therein. Other constructions based on the Suzuki curve and the curve with equation y q + y = x q r +1 can be found in [32] and [36]. More recently, AG Codes from the GK curve have been constructed in [1,3,9].In the present work we investigate one-point AG codes from the F q 2n -maximal GGS curve, n ≥ 5 odd. The GGS curve has more short orbits under its automorphism group than other maximal curves, see [15], and hence more possible structures for the Weierstrass semigroups at one point. On the one hand this makes the investigation more complicated; on the other hand it gives more chances of finding one-point AG codes with good parameters. One achievement of this work is the determination of the Weierstrass semigroup at any F q 2 -rational point.We show that the one-point codes at the infinite point P ∞ inherit a large automorphism group from the GGS curve; for many of such codes, the full automorphism group is obtained. Moreover, for q = 2, we compute explicitly the Feng-Rao designed minimum distance, which improves the Goppa designed minimum distance. As an application, we provide families of codes with q = 2 whose relative Singleton defect goes to zero as n goes to infinity. We were not able to produce analogous results for an F q 2 -rational affine point P 0 , because of the more complicated structure of the Weierstrass semigroup. In a comparison between onepoint codes from P ∞ and one-point codes from P 0 , it turns out that the best codes come sometimes from P ∞ , other times from P 0 ; we give evidence of this fact with...

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“…Let k = F n 6 . Let X be the curve over k constructed by Giulietti and Korchmáros in [11]; it is denoted C 3 in [13]. Let J = Aut(X ).…”

Section: D Unmixed Actionsmentioning

confidence: 99%

Automorphisms of Harbater–Katz–Gabber curves

et al. 2016

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Let k be a perfect field of characteristic p > 0, and let G be a finite group. We consider the pointed G-curves over k associated by Harbater, Katz, and Gabber to faithful actions of G on k[[t]] over k. We use such "HKG G-curves" to classify the automorphisms of k[[t]] of p-power order that can be expressed by particularly explicit formulas, namely those mapping t to a power series lying in a Z/pZ Artin-Schreier extension of k(t). In addition, we give necessary and sufficient criteria to decide when an HKG G-curve with an action of a larger finite group J is also an HKG J-curve.

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The automorphism groups of a family of maximal curves

Cited by 16 publications

References 18 publications

A complete characterization of Galois subfields of the generalized Giulietti–Korchmáros function field

A complete characterization of Galois subfields of the generalized Giulietti–Korchmáros function field

AG codes and AG quantum codes from the GGS curve

Automorphisms of Harbater–Katz–Gabber curves

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