We determine the full automorphism group of two recently constructed families S q andR q of maximal curves over finite fields. These curves are cyclic covers of the Suzuki and Ree curves, and are analogous to the Giulietti-Korchmáros cover of the Hermitian curve. We show thatS q is not Galois covered by the Hermitian curve maximal over F q 4 , andR q is not Galois covered by the Hermitian curve maximal over F q 6 . Finally, we compute the genera of many Galois subcovers ofS q andR q ; this provides new genera for maximal curves.Lemma 13. The lifted group LR(q) contains a subgroupR(q) isomorphic to the Ree group Aut(R q ).Lemma 14. The normalizer of C m in Aut(R q ) is the direct productR(q) × C m .Corollary 15. The group LR(q) coincides with the normalizer of C m in Aut(R q ).

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...

The Deligne-Lusztig curves associated to the algebraic groups of type 2 A 2 , 2 B 2 , and 2 G 2 are classical examples of maximal curves over finite fields. The Hermitian curve H q is maximal over F q 2 , for any prime power q, the Suzuki curve S q is maximal over F q 4 , for q = 2 2h+1 , h ≥ 1 and the Ree curve R q is maximal over F q 6 , for q = 3 2h+1 , h ≥ 0. In this paper we show that S 8 is not Galois covered by H 64 . We also give a proof for an unpublished result due to Rains and Zieve stating that R 3 is not Galois covered by H 27 . Furthermore, we determine the spectrum of genera of Galois subcovers of H 27 , and we point out that some Galois subcovers of R 3 are not Galois subcovers of H 27 . divide three times the order of any maximal subgroup of PSU(3, 64), a contradiction.Case |G| = 75. By Sylow and Schur-Zassenhaus theorems, G is a semidirect product G = S 5 ⋊ C 3 . By Theorem 2.7, ∆ = i · 65 + (24 − i) · 0 + j · 2 + (50 − j) · 3 with 0 ≤ i ≤ 24 and 0 ≤ j ≤ 50. This contradict (3.1).Case |G| = 78. By Sylow's Third Theorem, n 13 = 1; by Lemma 2.3, G acts on the fixed points of S 13 . Every nontrivial element σ ∈ S 13 generates S 13 and is either of type (A) or (B1). Hence, all nontrivial elements of G either are of type (A), or act on a common triangle T . In the former case, G contains a 2-element of type (A), contradicting Lemma 2.2. In the latter case, by the orbit-stabilizer theorem, the subgroup H of G fixing T pointwise contains a 2-element or a 3-element. This contradicts Lemma 2.2.Case |G| = 80. By [36, Theorem 1], G has a characteristic 2-subgroup N. By Lemma 2.3, G fixes the unique fixed point of N on H 64 , which is F 64 2 -rational. By Theorem 2.9,with v + w = 4, which is impossible.Case |G| = 84. By Sylow's Third Theorem, n 7 = 1. This contradicts Lemma 2.6.Case |G| = 90. Since |G| ≡ 2 (mod 4), G has a normal subgroup N of index 2 (see [31, Ex. 4.3]). By Sylow's Third Theorem, N has a characteristic 5-subgroup C 5 , so that C 5 is normal in G and n 5 = 1. Also, n 3 = 1. Then G is a semidirect product G = C 5 × S 3 ⋊ C 2 . By Theorem 2.7, ∆ = 4 · i + 40 · 2 + n 2 · 66 + (45 − n 2 ) · 1, with i ∈ {0, 65} and 1 < n 2 | 45. This contradicts (3.1).Case |G| = 91. By Theorem 2.7, ∆ = 78 · 2 + 12 · i with i ∈ {0, 65}, contradicting (3.1).Case |G| = 96. By [36, Theorem 1], G has a characteristic 2-subgroup N. By Lemma 2.3, G fixes the unique fixed point of N on H 64 , which is F 64 2 -rational. By Theorem 2.9,with v + w = 5, which is impossible.Case |G| = 100. By Sylow's Third Theorem, n 5 = 1. By Lemma 2.4, the fixed points of S 5 are the vertices of a triangle T . By Lemma 2.3, G acts on T . By the orbit-stabilizer theorem, G contains a 2-element fixing T pointwise. This contradicts Lemma 2.2.Case |G| = 104. By Sylow's Third Theorem, n 13 = 1. This contradicts Lemma 2.6.Case |G| = 105. By Sylow's Third Theorem, n 5 ∈ {1, 21}. All elements of a S 5 are of the same type, either (A) or (B1). Then, by Theorem 2.7, ∆ = 4i · 65 + 4(n 5 − i) · 0 + (104 − 4n 5 ) · 2, with 0 ≤ i ≤ n 5 . This contradic...

In this paper we investigate multi-point Algebraic-Geometric codes associated to the GK maximal curve, starting from a divisor which is invariant under a large automorphism group of the curve. We construct families of codes with large automorphism groups.

In this article we construct for any prime power q and odd n 5, a new F q 2n -maximal curve Xn. Like the Garcia-Güneri-Stichtenoth maximal curves, our curves generalize the Giulietti-Korchmáros maximal curve, though in a different way. We compute the full automorphism group of Xn, yielding that it has precisely q(q 2 − 1)(q n + 1) automorphisms. Further, we show that unless q = 2, the curve Xn is not a Galois subcover of the Hermitian curve. Finally, up to our knowledge, we find new values of the genus spectrum of F q 2n -maximal curves, by considering some Galois subcovers of Xn.