Goppa code and quantum stabilizer codes from plane curves given by separated polynomials
Vahid Nourozi, Farzaneh Ghanbari
TL;DR
The paper develops algebraic-geometry codes from plane curves defined by separated polynomials and leverages Hermitian self-orthogonality to construct quantum stabilizer codes with explicit parameter families. It defines AG codes on a curve $\\mathcal{X}$ over $\\mathbb{F}_{q^2}$ using divisors $D$ supported on $\\mathcal{X}(\\mathbb{F}_{q^2})$ and $G$ supported on $\\mathcal{X}(\\mathbb{F}_{q})$, deriving dual relations $C_r^{\\perp} = C_{q^3+q^2-3q-r}$ and self-orthogonality conditions $2r \\le q^3+q^2-3q$, with a key result that for $r \\le q^2+q-3$ the codes are Hermitian self-orthogonal. This enables a quantum code construction yielding $[[q^3, q^3+q^2-3q-2r, r+2q-q^2]]_q$ codes for $q^2-2 \\le r \\le q^2+q-3$, and the paper provides concrete examples at small field sizes demonstrating good code parameters. Overall, the work advances quantum coding by exploiting the arithmetic of plane curves given by separated polynomials to produce efficient stabilizer codes from AG codes.
Abstract
In this paper, we examine algebraic geometric (AG) codes associated with curves generated by separated polynomials, and we create AG codes and quantum stabilizer codes from these curves by varying their parameters. Our research involves a thorough examination of the curves' algebraic features as well as the creation of Goppa codes over them. Extending these findings, we create quantum stabilizer codes, revealing that quantum codes built from Hermitian self-orthogonal AG codes have acceptable parameters, improving the reliability and performance of communication networks.