Application of the Cartier Operator in Coding Theory

Abstract

The $a$-number is an invariant of the isomorphism class of the $p$-torsion group scheme. We use the Cartier operator on $H^0({\mathcal{A}}_2,{\Omega }^1)$ to find a closed formula for the $a$-number of the form ${\mathcal{A}}_2=v(Y^{\sqrt{q}}+Y-x^{\frac{\sqrt{q}+1}{2}})$ where $q=p^s$ over the finite field ${\mathbb{F}}_{q^2}$. The application of the computed $a$-number in coding theory is illustrated by the relationship between the algebraic properties of the curve and the parameters of codes that are supported by it.