Application of the Cartier Operator in Coding Theory
Vahid Nourozi
TL;DR
The paper determines the $a$-number of the maximal Artin-Schreier curve ${\mathcal A}_2$ over $\mathbb{F}_{q^2}$ by applying the Cartier operator to a basis of holomorphic differentials, obtaining an explicit closed form $a({\mathcal A}_2)=\frac{p-1}{8}\big(\sqrt{p^{s-2}}+1\big)(\sqrt{q}-1)$ for even $s$. The method hinges on representing holomorphic differentials, leveraging the formula $\mathfrak{C}\big(h\frac{dx}{F_y}\big)=(\nabla(F^{p-1}h))^{1/p}\frac{dx}{F_y}$ and a rank analysis of the induced $p$-th power Cartieir matrix, with base cases and induction guiding the general result. A concrete MAGMA example illustrates the computation. The paper then connects this invariant to coding theory by showing how the $a$-number yields a lower bound $\delta\ge d-m-2g+a(X)$ for codes $C(D,G)$ constructed via the Cartier operator, highlighting the practical impact on minimum distance, code design, and error correction for algebraic-geometry codes based on ${\mathcal A}_2$.
Abstract
The $a$-number is an invariant of the isomorphism class of the $p$-torsion group scheme. We use the Cartier operator on $H^0(\mathcal{A}_2,Ω^1)$ 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.