Euclidean and Hermitian self-orthogonal algebraic geometry codes and their application to quantum codes

• Main Authors: Jin, Lingfei, Xing, Chaoping
• Other Authors: School of Physical and Mathematical Sciences
• Format: Article
• Language: English
• Published: 2013
• Online Access: https://hdl.handle.net/10356/95846; http://hdl.handle.net/10220/11432
• Institution: Nanyang Technological University

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.