Weierstrass Semigroups and Codes from a Quotient of the Hermitian Curve

Abstract

We consider the quotient of the Hermitian curve defined by the equation y q + y = x m over F q 2 where m > 2 is a divisor of q + 1. For 2 ≤ r ≤ q + 1, we determine the Weierstrass semigroup of any r-tuple of F q 2 -rational points (P ∞ , P 0b 2 , . . . , P 0br ) on this curve. Using these semigroups, we construct algebraic geometry codes with minimum distance exceeding the designed distance. In addition, we prove that there are r-point codes, that is codes of the form C (D, α 1 P ∞ , α 2 P 0b 2 , + · · · + α r P 0br ) where r ≥ 2, with better parameters than any comparable one-point code on the same curve. Some of these codes have better parameters than comparable one-point Hermitian codes over the same field. All of our results apply to the Hermitian curve itself which is obtained by taking m = q + 1 in the above equation.

On Weierstrass semigroups and sets: a review with new results

“…. , P m are collinear-see also [24] for the determination of H (P m ) for some quotient of Hermitian curves. In [23] we find the determination of H (P 1 , P 2 ) for a pair of points on the Suzuki curve and in [20] we find the determination of all Weierstrass semigroups at a pair of Weierstrass points whose first nongaps are three (here the field is assumed to have characteristic zero).…”

Weierstrass Semigroup and Pure Gaps at Several Points on the GK Curve

“…For this curve, taking m = 1, by Theorem 3.4, we have that (2,11), (3,3), (4, 13), (5, 5), (7, 7), (10, 10), (11,2), (13,4), (19, 1)} .…”

Pure Weierstrass gaps from a quotient of the Hermitian curve

“…Matthews [14] investigated the Weierstrass semigroup of any collinear places on a Hermitian curve. In [15], Matthews generalized the results of [7], [14] by determining the Weierstrass semigroup of arbitrary rational places on the quotient of the Hermitian curve defined by the equation y m = x q + x over F q 2 where q is a prime power and m > 2 is a divisor of q + 1. For general Kummer extensions, the authors in [16], [9] recently described the Weierstrass semigroups and gaps at one place and two places.…”