Entanglement-assisted quantum error correcting codes (EAQECCs) play an important role against errors in quantum information system. Especially maximal entanglement EAQECCs can achieve a higher rate and have better noise suppression capabilities. In this paper, LCD caps inPG(3, 9) andPG(4, 9) are discussed and used to construct maximal entanglement EAQECCs with distance 4. According to Hamming bound, most of these results are optimal.
Entanglement-assisted quantum error correcting codes (EAQECCs) play an important role against errors in quantum information system. Especially maximal entanglement EAQECCs can achieve a higher rate and have better noise suppression capabilities. In this paper, LCD caps inPG(3, 9) andPG(4, 9) are discussed and used to construct maximal entanglement EAQECCs with distance 4. According to Hamming bound, most of these results are optimal.
The entanglement-assisted (EA) formalism is a generalization of the standard stabilizer formalism, and it can transform arbitrary quaternary classical linear codes into entanglement-assisted quantum error correcting codes (EAQECCs) by using of shared entanglement between the sender and the receiver. Using a decomposition of the 126-cap in PG(5,4), we firstly constructed LCD n-cap with 6The entanglement-assisted (EA) formalism is a generalization of the standard stabilizer formalism, and it can transform arbitrary quaternary classical linear codes into entanglement-assisted quantum error correcting codes (EAQECCs) by using of shared entanglement between the sender and the receiver. Using a decomposition of the 126-cap in PG(5,4), we firstly constructed LCD n-cap with 6≤n≤120. From these LCD caps, we then derived the related maximal entanglement [[n, n-6,4;6]] EAQECCs. Finally, using LCD LCD subcaps of 126-cap obtained, we constructed maximal entanglement EAQECCs with parameters [[n, n-;k,4;k]] for 6≤k≤11.
Maximal entanglement entanglement-assisted quantum error correcting codes (EAQECCs) can achieve the EA-hashing bound asymptotically, and a higher rate and/or better noise suppression capabilities may be achieved by exploiting maximal entanglement. In this work, we discuss construction of some families of [n, k, d] zero radical codes with minimal distance d ≥ 6, 8, 10, 12, 14 from two classes of quaternary Bose-Chaudhuri-Hocquenghem (BCH) codes. Using these zero radical codes, we construct many maximal entanglement EAQECCs with very good parameters. We calculate the parameters of these EAQECCs by easily algebraical method, rather than by usually method of computation by computer. The maximal entanglement EAQECCs presented here have better parameters than those available in the literature.
Entanglement-assisted quantum error correcting code (EAQECC) is a generalization of standard stabilizer quantum code. Maximal entanglement EAQECCs can achieve the EA-hashing bound asymptotically. In this work, we give elementary recursive constructions of quaternary zero radical codes with dual distance three for all n ≥ 4. Consequently, good maximal entanglement EAQECCs of minimum distance three for such length n are obtained. Almost all of these EAQECCs are optimal or near optimal according to the EA-quantum Hamming bound.
Entanglement-assisted quantum error-correcting codes (EAQECCs) make use of preexisting entanglement between the sender and receiver to boost the rate of transmission. It is possible to construct an EAQECC from any classical linear code, unlike standard quantum error-correcting codes, which can only be constructed from dual-containing codes. However, the parameter of ebits c is usually calculated by computer search. In this paper, we construct four classes of MDS entanglement-assisted quantum error-correcting codes (MDS EAQECCs) based on k-Galois LCD MDS codes for some certain code lengths, where the parameter of ebits c can be easily generated algebraically and not by computational search. Moreover, the constructed four classes of EAQECCs are also maximal-entanglement EAQECCs.
Hermitian linear complementary dual (LCD) codes are a class of linear codes that intersect with their Hermitian dual trivially. Each Hermitian LCD code can give an entanglement-assisted quantum error-correcting code (EAQECC) with maximal entanglement. Methods of constructing Hermitian LCD codes from known codes were developed, and seven new Hermitian LCD codes with parameters $ [119,4,88]_{4} $, $ [123,4,91]_{4} $, $ [124,4,92]_{4} $, $ [136,4,101]_{4} $, $ [140,4,104]_{4} $, $ [188,4,140]_{4} $ and $ [212,4,158]_{4} $ were constructed. Seven families of Hermitian LCD codes and their related EAQECCs were derived from these codes. These new EAQECCs have better parameters than those known in the literature.
The classical linear codes that construct quantum error correction codes need to meet certain specific conditions. The proposal of entanglement-assisted quantum error correcting codes eliminates the restriction of these conditions, so all classical linear codes can be constructed into entanglement-assisted quantum error correcting codes. The Hadamard matrix is a special type of orthogonal matrix. Hadamard codes, constructed based on the standard form of Hadamard matrices, are a type of linear complementary dual (LCD) codes. By extending the generator and parity-check matrices of Hadamard codes, two types of classical codes whose parameters are [(l+1)(n−1), n−1] and [3n−3, n−1] maintain the LCD property. Utilizing the relationship between their LCD property and the dimensions of the Euclidean hull, this paper constructs two types of EAQECCs based on Hadamard matrices. These EAQECCs can achieve maximal entanglement, resulting in higher code rates and net rates compared to other EAQECCs. Their parameters are [[3n−3, 2n−2, d≥1; n−1]] and [[2n−2, n−1, d≥1; n−1]], respectively.