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.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.

MathematicsSep 3, 20208 citations

Optimal, Best-Known Divisible Binary Codes and Related Quantum Codes

Hao Chen, Cong Yu, Yaqi Chen +3 more

IEEE Transactions on Information TheoryAug 4, 20260 citations

Binary Construction of Quantum Codes of Minimum Distance Three and Four

Ruihu Li, Xinran Li

We give elementary recursive constructions of binary self-orthogonal codes with dual distance four for all even lengths n/spl ges/12 and n=8. Consequently, good quantum codes of minimum distance three and four for such length n are obtained via Steane's construction and the CSS construction. Previously, such quantum codes were explicitly constructed only for a sparse set of lengths. Almost all of our quantum codes of minimum distance three are optimal or near optimal, and some of our minimum-distance four quantum codes are better than or comparable with those known before.We give elementary recursive constructions of binary self-orthogonal codes with dual distance four for all even lengths n/spl ges/12 and n=8. Consequently, good quantum codes of minimum distance three and four for such length n are obtained via Steane's construction and the CSS construction. Previously, such quantum codes were explicitly constructed only for a sparse set of lengths. Almost all of our quantum codes of minimum distance three are optimal or near optimal, and some of our minimum-distance four quantum codes are better than or comparable with those known before.

IEEE Transactions on Information TheoryJun 1, 200467 citations

New Binary Quantum Codes Constructed from Quasi-Cyclic Codes

Chaofeng Guan, Ruihu Li, Liangdong Lü +1 more

arXiv:2112.07137International Journal of Theoretical PhysicsJun 1, 202212 citations

Generalized Binary Representation for the Nonbinary LDPC Code With Decoder Design

Yang Yu, Wen Chen, Jun Li +2 more

In this paper, we consider the performance-optimized nonbinary low-density parity check code over general linear group, i.e., $\bar{\cal C}$ . A new methodology for constructing the binary representation [generalized binary representation (GBR)] of $\bar{\cal C}$ is proposed, which can be optimized with regard to both degree distributions and girth. As to the decoding of the GBR, we develop a low-complexity hybrid parallel decoding process. It is shown that the decoding performance of the GBR under the proposed binary decoding process could closely approach the decoding performance of its mother code $\bar{\cal C}$ under nonbinary belief propagation decoding. A simple code optimization algorithm for the GBR is also provided. Simulations show the comparative results and justify the advantages of the proposed constructions.In this paper, we consider the performance-optimized nonbinary low-density parity check code over general linear group, i.e.,$\bar{\cal C}$. A new methodology for constructing the binary representation [generalized binary representation (GBR)] of$\bar{\cal C}$is proposed, which can be optimized with regard to both degree distributions and girth. As to the decoding of the GBR, we develop a low-complexity hybrid parallel decoding process. It is shown that the decoding performance of the GBR under the proposed binary decoding process could closely approach the decoding performance of its mother code$\bar{\cal C}$under nonbinary belief propagation decoding. A simple code optimization algorithm for the GBR is also provided. Simulations show the comparative results and justify the advantages of the proposed constructions.

IEEE Transactions on CommunicationsJul 30, 20146 citations

In this paper, we consider the performance-optimized nonbinary low-density parity check code over general linear group, i.e.,

\bar{\cal C}

. A new methodology for constructing the binary representation [generalized binary representation (GBR)] of

\bar{\cal C}

is proposed, which can be optimized with regard to both degree distributions and girth. As to the decoding of the GBR, we develop a low-complexity hybrid parallel decoding process. It is shown that the decoding performance of the GBR under the proposed binary decoding process could closely approach the decoding performance of its mother code

\bar{\cal C}

under nonbinary belief propagation decoding. A simple code optimization algorithm for the GBR is also provided. Simulations show the comparative results and justify the advantages of the proposed constructions.