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....
| Main Authors: | , |
|---|---|
| Other Authors: | |
| Format: | Journal Article |
| Language: | English |
| Published: |
2013
|
| Online Access: | https://hdl.handle.net/10356/95846 http://hdl.handle.net/10220/11432 |
| _version_ | 1811678886732234752 |
|---|---|
| author | Jin, Lingfei Xing, Chaoping |
| author2 | School of Physical and Mathematical Sciences |
| author_facet | School of Physical and Mathematical Sciences Jin, Lingfei Xing, Chaoping |
| author_sort | Jin, Lingfei |
| collection | NTU |
| description | 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. |
| first_indexed | 2024-10-01T03:00:23Z |
| format | Journal Article |
| id | ntu-10356/95846 |
| institution | Nanyang Technological University |
| language | English |
| last_indexed | 2024-10-01T03:00:23Z |
| publishDate | 2013 |
| record_format | dspace |
| spelling | ntu-10356/958462020-03-07T12:37:21Z Euclidean and Hermitian self-orthogonal algebraic geometry codes and their application to quantum codes Jin, Lingfei Xing, Chaoping School of Physical and Mathematical Sciences 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. 2013-07-15T06:54:40Z 2019-12-06T19:22:17Z 2013-07-15T06:54:40Z 2019-12-06T19:22:17Z 2011 2011 Journal Article Jin, L., & Xing, C. (2012). Euclidean and Hermitian Self-Orthogonal Algebraic Geometry Codes and Their Application to Quantum Codes. IEEE Transactions on Information Theory, 58(8), 5484-5489. https://hdl.handle.net/10356/95846 http://hdl.handle.net/10220/11432 10.1109/TIT.2011.2177066 en IEEE transactions on information theory © 2011 IEEE. |
| spellingShingle | Jin, Lingfei Xing, Chaoping Euclidean and Hermitian self-orthogonal algebraic geometry codes and their application to quantum codes |
| title | Euclidean and Hermitian self-orthogonal algebraic geometry codes and their application to quantum codes |
| title_full | Euclidean and Hermitian self-orthogonal algebraic geometry codes and their application to quantum codes |
| title_fullStr | Euclidean and Hermitian self-orthogonal algebraic geometry codes and their application to quantum codes |
| title_full_unstemmed | Euclidean and Hermitian self-orthogonal algebraic geometry codes and their application to quantum codes |
| title_short | Euclidean and Hermitian self-orthogonal algebraic geometry codes and their application to quantum codes |
| title_sort | euclidean and hermitian self orthogonal algebraic geometry codes and their application to quantum codes |
| url | https://hdl.handle.net/10356/95846 http://hdl.handle.net/10220/11432 |
| work_keys_str_mv | AT jinlingfei euclideanandhermitianselforthogonalalgebraicgeometrycodesandtheirapplicationtoquantumcodes AT xingchaoping euclideanandhermitianselforthogonalalgebraicgeometrycodesandtheirapplicationtoquantumcodes |