Algebraic Curves over a Finite Field

3-Nets realizing a group in a projective plane

“…1. [7,Theorem 6.104] A nonsingular plane cubic F can be equipped with an additive group (F, +) on the set of all its points. If an inflection point P 0 of F is chosen to be the identity 0, then three distinct points P , Q, R ∈ F are collinear if and only if P + Q + R = 0.…”

Special divisors of large dimension on curves with many points over finite fields

“…We have the Castelnuovo bound ( [4], Theorem 7.111, [5], Theorem 2.9) which states that, if Y is a curve embedded in P n of degree k, not contained in a hyperplane, not strange, and k−1 = m(n−1)+r, 0 ≤ r ≤ n − 2, then the genus of Y is at most m(m − 1)(n − 1)/2 + mr. Applying this bound to to X ⊂ P s as in the lemma proves the corollary.…”

“…Their work was motivatived by an application to the construction of binary error correcting codes with good properties by concatenating codes from large alphabets (such as algebraic geometry codes) with a Hadamard code. They remarked that, if divisors with certain properties could be constructed on curves meeting the Drinfeld-Vladut bound ( [4], Theorem 9.37) their construction could be improved. Unfortunately, we show in this note that such divisors do not exist.…”

Frobenius nonclassicality with respect to linear systems of curves of arbitrary degree

“…. , d − 1}, the curve X has a linear series D s of dimension M = s+2 2 − 1 and degree sd [11,Section 7.7]. Applying Stöhr-Voloch's theorem [17 The bound (1.3) improves the Hasse-Weil bound in several cases ( [17, section 3], [6]).…”