A Generic Construction on Self-orthogonal Algebraic Geometric Codes and Its Applications

TL;DR

This paper introduces a generalized criterion for self-orthogonality in algebraic geometric codes, enabling the construction of self-dual and almost self-dual codes with applications in quantum error correction.

Contribution

It presents a new criterion based on residues for characterizing self-orthogonality and a generic construction method from self-dual MDS codes for AG codes.

Findings

Constructed families of self-dual and almost self-dual AG codes.

Codes achieve parameters close to the Singleton bound.

Some codes are applicable to quantum code construction.

Abstract

In the realm of algebraic geometric (AG) codes, characterizing dual codes has long been a challenging task. In this paper we introduces a generalized criterion to characterize self-orthogonality of AG codes based on residues, drawing upon the rich algebraic structures of finite fields and the geometric properties of algebraic curves. We also present a generic construction of self-orthogonal AG codes from self-dual MDS codes. Using these approaches, we construct several families of self-dual and almost self-dual AG codes. These codes combine two merits: good performance as AG code whose parameters are close to the Singleton bound together with Euclidean (or Hermtian) self-dual/self-orthogonal property. Furthermore, some AG codes with Hermitian self-orthogonality can be applied to construct quantum codes with notably good parameters.