On Galois self-orthogonal algebraic geometry codes

TL;DR

This paper introduces new classes of Galois self-orthogonal algebraic geometry codes, providing criteria for their construction and expanding the known MDS codes from various algebraic curves.

Contribution

It presents a criterion for Galois SO AG codes and constructs new MDS Galois SO codes from projective lines, elliptic, hyper-elliptic, and Hermitian curves.

Findings

New classes of MDS Galois SO AG codes constructed from algebraic curves.

A criterion for identifying Galois SO AG codes.

An embedding method to generate more MDS Galois SO codes.

Abstract

Galois self-orthogonal (SO) codes are generalizations of Euclidean and Hermitian SO codes. Algebraic geometry (AG) codes are the first known class of linear codes exceeding the Gilbert-Varshamov bound. Both of them have attracted much attention for their rich algebraic structures and wide applications in these years. In this paper, we consider them together and study Galois SO AG codes. A criterion for an AG code being Galois SO is presented. Based on this criterion, we construct several new classes of maximum distance separable (MDS) Galois SO AG codes from projective lines and several new classes of Galois SO AG codes from projective elliptic curves, hyper-elliptic curves and hermitian curves. In addition, we give an embedding method that allows us to obtain more MDS Galois SO codes from known MDS Galois SO AG codes.