On some Galois covers of the Suzuki and Ree curves

Abstract

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 ).

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

“…• AG convolutional codes (Section 7); To this aim, some geometrical information onR q andS q is needed, which is collected in Section 2.4. After recalling some known facts from the literature ( [16,43]), we also prove that forR q andS q the Weierstrass semigroup at the unique infinite place is symmetric, and we deduce that the F q -rational places are Weierstrass points. We also provide new plane models both forR q andS q .…”

“…• A cyclic group C m generated by the automorphism τ : (x, y, t) → (x, y, λt), where λ ∈ F q 4 is a primitive m-th root of unity; C m is the Galois group of the coverS q → S q . The full automorphism group Aut(S q ) ofS q was computed in [16] and is a direct product S(q) × C m , whereS(q) ∼ = S(q). Also, Aut(S q ) has exactly two short orbits: one short orbit O 1 has size q 2 + 1 and coincides withS q (F q ); the other short orbit O 2 has size |S(q)|, and hence the stabilizer in Aut(S q ) of a place in O 2 has order m. The contribution to the different divisor of every element in Aut(S q ) is also described, as summarized in the following lemma.…”

“…By Theorem 2.5, Condition C1) is satisfied. From[16, Proposition 25],T := {(x, y, t) → (x + b, y + b q 0 x + c, t) | b, c ∈ F q }…”

Quotients of the Hermitian curve from subgroups of $$\mathrm{PGU}(3,q)$$ without fixed points or triangles

“…New genera of F q 2 -maximal curves. The results of Section 4 provide new genera of maximal curves over finite fields, with respect to the genera obtained in [1,2,3,7,8,9,10,12,15,16,17,18,26,28,30]. Table 2 collects some examples.…”

Classification of all Galois subcovers of the Skabelund maximal curves

“…The curve Sq is maximal over the field F q 4 . Its automorphism group is Aut( Sq ) = Sz(q) × C m , see [30].…”