  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.
