On Galois self-orthogonal algebraic geometry codes
Yun Ding, Shixin Zhu, Xiaoshan Kai, Yang Li
TL;DR
The paper addresses the existence of general Galois self-orthogonal ($e$-Galois SO) AG codes by establishing a criterion that guarantees an AG code is $e$-Galois SO. It develops an embedding method to derive additional MDS $e$-Galois SO AG codes from line-based codes and provides explicit constructions over projective lines as well as elliptic, hyper-elliptic, and Hermitian curves, yielding multiple new MDS or near-MDS codes. The work broadens the scope of Galois SO codes beyond Euclidean/Hermitian cases, delivering concrete parameter families and demonstrating potential for applications in areas like entanglement-assisted quantum coding. Overall, it significantly expands the catalog of AG-based Galois SO codes and lays groundwork for further exploration of Galois hulls in AG 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.