On the curve <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msup><mml:mrow><mml:mi>y</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:mi>x</mml:mi></mml:math> over finite fields

Motivated by previous computations in Garcia, Stichtenoth and Xing (2000) paper [8], we discuss the spectrum M(q 2 ) for the genera of maximal curves over finite fields of order q 2 with 7 ≤ q ≤ 16. In particular, by using a result in Kudo and Harashita (2016) paper [16], the set M(7 2 ) is completely determined.Let r be the dimension of D. Then r ≥ 2 by (2.2), and the condition r = 2 is equivalent to g = q(q−1)/2, or equivalent to X being K-isomorphic to the Hermitian curve y q+1 = x q +x [23], [6]. Under certain conditions, we have a similar result for r = 3 in Corollary 2.3 and Proposition 3.1. In fact, in Section 3 we bound g via Stöhr-Voloch theory [20]

show abstract

“…Applying Kleiman-Serre covering result to these curves and to their generalizations [30], [7] and [71] provided further examples of maximal curves; see [1,8,15,22,38,39]. Other recent constructions can be found in [42,[78][79][80][81].…”

Section: Introductionmentioning

confidence: 97%