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 ${\mathbb F}_{q^2}$ where m > 2 is a divisor of q+1. For 2≤ r ≤ q+1, we determine the Weierstrass semigroup of any r-tuple of ${\mathbb F}_{q^2}$-rational points $(P_\infty, P_{0b_2},\ldots,P_{0b_r})$ 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 $C_\Omega(D, \alpha_1P_\infty, \alpha_2P_{0b_2},+\cdots+ \alpha_rP_{0b_r})$ 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