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International Journal of Quantum Information

Entanglement-assisted quantum codes constructed from primitive quaternary BCH codes

Liangdong Lü, Ruihu Li

Published Apr 1, 2014• 45 citations
Publisher

Abstract

The entanglement-assisted (EA) formalism generalizes the standard stabilizer formalism. All quaternary linear codes can be transformed into entanglement-assisted quantum error correcting codes (EAQECCs) under this formalism. In this work, we discuss construction of EAQECCs from Hermitian non-dual containing primitive Bose–Chaudhuri–Hocquenghem (BCH) codes over the Galois field GF(4). By a careful analysis of the cyclotomic cosets contained in the defining set of a given BCH code, we can determine the optimal number of ebits that needed for constructing EAQECC from this BCH code, rather than calculate the optimal number of ebits from its parity check matrix, and derive a formula for the dimension of this BCH code. These results make it possible to specify parameters of the obtained EAQECCs in terms of the design parameters of BCH codes.

DOI
10.1142/s0219749914500154

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The entanglement-assisted (EA) formalism generalizes the standard stabilizer formalism. All quaternary linear codes can be transformed into entanglement-assisted quantum error correcting codes (EAQECCs) under this formalism. In this work, we discuss construction of EAQECCs from Hermitian non-dual containing primitive Bose–Chaudhuri–Hocquenghem (BCH) codes over the Galois field GF(4). By a careful analysis of the cyclotomic cosets contained in the defining set of a given BCH code, we can determine the optimal number of ebits that needed for constructing EAQECC from this BCH code, rather than calculate the optimal number of ebits from its parity check matrix, and derive a formula for the dimension of this BCH code. These results make it possible to specify parameters of the obtained EAQECCs in terms of the design parameters of BCH codes.
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.
Quantum error correcting codes (QECCs) play an important role in preventing quantum information decoherence. Good quantum stabilizer codes were constructed by classical error correcting codes. In this paper, Bose-Chaudhuri-Hocquenghem (BCH) codes over finite fields are used to construct quantum codes. First, we try to find such classical BCH codes, which contain their dual codes, by studying the suitable cyclotomic cosets. Then, we construct nonbinary quantum BCH codes with given parameter sets. Finally, a new family of quantum BCH codes can be realized by Steane's enlargement of nonbinary Calderbank-Shor-Steane (CSS) construction and Hermitian construction. We have proven that the cyclotomic cosets are good tools to study quantum BCH codes. The defining sets contain the highest numbers of consecutive integers. Compared with the results in the references, the new quantum BCH codes have better code parameters without restrictions and better lower bounds on minimum distances. What is more, the new quantum codes can be constructed over any finite fields, which enlarges the range of quantum BCH codes.
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 ccc 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 kkk-Galois LCD MDS codes for some certain code lengths, where the parameter of ebits ccc can be easily generated algebraically and not by computational search. Moreover, the constructed four classes of EAQECCs are also maximal-entanglement EAQECCs.
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 the special structure of linear EAQECCs, we derive an EA-Plotkin bound for linear EAQECCs, which strengthens the previous known EA-Plotkin bound. This linear EA-Plotkin bound is tighter then the EA-Singleton bound, and matches the EA-Hamming bound and the EA-linear programming bound in some cases. We also construct three families of EAQECCs with good parameters. Some of these EAQECCs saturate this linear EA-Plotkin bound and the others are near optimal according to this bound; almost all of these linear EAQECCs are degenerate codes.
The entanglement-assisted (EA) formalism is a generalization of the standard stabilizer formalism, and it can transform classical linear quaternary codes into entanglement-assisted quantum error correcting codes (EAQECCs) by using of shared entangled qubits between the sender and the receiver.In this work, we give elementary recursive constructions of special quaternary codes of length n and dual distance four that constructed from known caps in projective space PG(5,4) and PG(6,4) for all length 6n283.Consequently, good maximal entanglement EAQECCs of minimum distance four for such length n are constructed from the obtained quaternary codes.
Classical Bose-Chaudhuri-Hocquenghem (BCH) codes over finite fields have been studied extensively. One can construct quantum stabilizer codes with good parameters using classical BCH codes. In this paper, our goal is to find such classical BCH codes. We study some properties of suitable cyclotomic cosets at first. These results make it possible to construct nonbinary quantum BCH codes with a given parameter set. Several new families of quantum BCH codes obtained are based on Steane's enlargement of nonbinary Calderbank-Shor-Steane codes and Hermitian construction, respectively. Meanwhile, we have shown that the cyclotomic cosets given in our schemes are optimal to design quantum BCH codes. The defining set contains most consecutive integers. Therefore, corresponding quantum stabilizer codes have better lower bound of minimum distance. Furthermore, it is convenient to compute the dimension of new quantum codes. Compared with the ones available in the literature, the quantum BCH codes in our schemes have good parameters. In particular, we extend known results to more general case.