Abstract
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.
Highlights
Quantum error-correcting codes have attracted wide attention in recent years due to their applications in quantum communications and quantum computations [1,2,3]
CSS codes (Calderbank-Sho-Steane) 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
As we can see from the CSS construction of quantum codes [1,3], self-orthogonal codes have been used for construction of quantum codes
Summary
Quantum error-correcting codes have attracted wide attention in recent years due to their applications in quantum communications and quantum computations [1,2,3]. Wu and Lee [15] first used the difference of simplicial complexes for construction of binary linear complementary dual codes and binary self-orthogonal codes. (3) We obtain four infinite families of binary quantum codes for which the minimum distances are three or four; this is achieved by construction of binary self-orthogonal codes (Theorem 4). (4) we obtain infinite families of binary quantum codes using these self-orthogonal codes, which include all binary quantum Hamming codes with length n ≥ 7; some binary quantum codes with minimum distance three or four are obtained by using Theorem 4, and we confirm the optimality of the codes in the tables according to the database of Grassl [16].
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