Euclidean and Hermitian Self-orthogonal Algebraic Geometry Codes and Their Application to Quantum Codes

TL;DR

This paper demonstrates that algebraic geometry codes over finite fields of even characteristic are equivalent to Euclidean or Hermitian self-orthogonal codes when their dimension is just under half their length, enabling the construction of quantum codes with favorable asymptotic bounds.

Contribution

It establishes new conditions under which algebraic geometry codes are self-orthogonal, simplifying their use in quantum code construction without requiring complex differential existence proofs.

Findings

Algebraic geometry codes with dimension slightly less than half their length are equivalent to self-orthogonal codes.

The results apply to both Euclidean and Hermitian self-orthogonal algebraic geometry codes.

Application to quantum codes yields asymptotically good quantum codes.

Abstract

In the present paper, we show that if the dimension of an arbitrary algebraic geometry code over a finite field of even characters is slightly less than half of its length, then it is equivalent to an Euclidean self-orthogonal code. However, in the literatures, a strong contrition about existence of certain differential is required to obtain such a result. We also show a similar result on Hermitian self-orthogonal algebraic geometry codes. As a consequence, we can apply our result to quantum codes and obtain quantum codes with good asymptotic bounds.