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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.

 

 

 

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