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.

翻译：伽罗瓦自正交（SO）码是欧几里得自正交码与埃尔米特自正交码的推广。代数几何（AG）码是首个已知超过吉尔伯特-瓦尔沙莫夫界的线性码类。这两类码因其丰富的代数结构及广泛应用而备受关注。本文将其结合研究，探讨伽罗瓦自正交代数几何码。我们给出了代数几何码成为伽罗瓦自正交码的判定准则。基于该准则，从射影直线构造了几类新的极大距离可分（MDS）伽罗瓦自正交代数几何码，并从射影椭圆曲线、超椭圆曲线及埃尔米特曲线构造了几类新的伽罗瓦自正交代数几何码。此外，我们提出一种嵌入方法，可通过已知的MDS伽罗瓦自正交代数几何码获得更多MDS伽罗瓦自正交码。