It is an important issue to search for self-orthogonal codes for construction of quantum codes by CSS construction (Calderbank-Sho-Steane codes); in quantum error correction, CSS codes are a special type of stabilizer codes constructed from classical codes with some special properties, and the CSS construction of quantum codes is a well-known construction. First, we employ hierarchical posets with two levels for construction of binary linear codes. Second, we find some necessary and sufficient conditions for these linear codes constructed using posets to be self-orthogonal, and we use these self-orthogonal codes for obtaining binary quantum codes. Finally, we obtain four infinite families of binary quantum codes for which the minimum distances are three or four by CSS construction, which include binary quantum Hamming codes with length n ≥ 7. We also find some (almost) "optimal" quantum codes according to the current database of Grassl. Furthermore, we explicitly determine the weight distributions of these linear codes constructed using posets, and we present two infinite families of some optimal binary linear codes with respect to the Griesmer bound and a class of binary Hamming codes.
Bibliographical noteFunding Information:
Y. Lee is supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST)(NRF-2017R1A2B2004574) and by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (grant No. 2019R1A6A1A11051177). Acknowledgments: We thank the reviewers of this paper for their helpful comments, which improved the clarity of this paper.
© 2020 by the authors.
- Binary linear code
- Quantum code
- Self-orthogonal code
- Weight distribution