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

In the present paper, we show that if the dimension of an arbitrary algebraic geometry code over a finite field of even characteristic is slightly less than n/2-g with n being the length of the code and g being the genus of the base curve, then it is equivalent to an Euclidean self-orthogonal code. Previously, such results required a strong condition on the existence of a certain differential. 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 some good quantum codes. In particular, we obtain a q-ary quantum [[q+1,1]]-MDS code for an even power q which is essential for quantum secret sharing.