Self-orthogonal and self-dual codes from maximal curves

TL;DR

This paper constructs self-orthogonal and self-dual algebraic geometric codes with parameters close to optimal, using algebraic and geometric properties, and also develops quantum codes with large minimum distances.

Contribution

It introduces new methods to construct self-orthogonal and self-dual codes from maximal curves, advancing algebraic geometric code theory.

Findings

Constructed self-orthogonal and self-dual codes with parameters close to the Singleton bound.

Developed quantum codes with large minimum distances.

Provided descriptions of differentials using algebraic structures and geometric properties.

Abstract

In the field of algebraic geometric codes (AG codes), the characterization of dual codes has long been a challenging problem which relies on differentials. In this paper, we provide some descriptions for certain differentials utilizing algebraic structure of finite fields and geometric properties of algebraic curves. Moreover, we construct self-orthogonal and self-dual codes with parameters $[n, k, d]_{q^2}$ satisfying $k + d$ is close to $n$. Additionally, quantum codes with large minimum distance are also constructed.