Linear $\ell$-Intersection Pairs of Codes and Their Applications

2019

In this paper, a linear l-intersection pair of codes is introduced as a generalization of linear complementary pairs of codes. Two linear codes are said to be a linear l-intersection pair if their intersection has dimension l. Characterizations and constructions of such pairs of codes are given in terms of the corresponding generator and parity-check matrices. Linear l-intersection pairs of MDS codes over Fq of length up to q + 1 are given for all possible parameters. As an application, linear l-intersection pairs of codes are used to construct entanglement-assisted quantum error correcting codes. This provides a large number of new MDS entanglement-assisted quantum error correcting codes.

Designs, Codes and Cryptography

In this paper, a linear ℓ-intersection pair of codes is introduced as a generalization of linear complementary pairs of codes. Two linear codes are said to be a linear ℓ-intersection pair if their intersection has dimension ℓ. Characterizations and constructions of such pairs of codes are given in terms of the corresponding generator and parity-check matrices. Linear ℓ-intersection pairs of MDS codes over F q of length up to q + 1 are given for all possible parameters. As an application, linear ℓ-intersection pairs of codes are used to construct entanglement-assisted quantum error correcting codes. This provides a large number of new MDS entanglement-assisted quantum error correcting codes.

2018

In this paper, a linear l-intersection pair of codes is introduced as a generalization of linear complementary pairs of codes. Two linear codes are said to be a linear lintersection pair if their intersection has dimension l. A characterization of such pairs of codes is given in terms of the corresponding generator and parity-check matrices. Linear l-intersection pairs of MDS codes over Fq of length up to q + 1 are given for all possible parameters. As an application, linear l-intersection pairs of codes are used to construct entanglement-assisted quantum error correcting codes. This provides a large number of new MDS entanglement-assisted quantum error correcting codes.

Designs, Codes and Cryptography, 2017

Entanglement-assisted quantum error correcting codes (EAQECCs) are a simple and fundamental class of codes. They allow for the construction of quantum codes from classical codes by relaxing the duality condition and using pre-shared entanglement between the sender and receiver. However, in general it is not easy to determine the number of shared pairs required to construct an EAQECC. In this paper, we show that this number is related to the hull of the classical code. Using this fact, we give methods to construct EAQECCs requiring desirable amount of entanglement. This leads to design families of EAQECCs with good error performance. Moreover, we construct maximal entanglement EAQECCs from LCD codes. Finally, we prove the existence of asymptotically good EAQECCs in the odd characteristic case.

IEEE Transactions on Information Theory, 2000

As in classical coding theory, quantum analogues of low-density parity-check (LDPC) codes have offered good error correction performance and low decoding complexity by employing the Calderbank-Shor-Steane (CSS) construction. However, special requirements in the quantum setting severely limit the structures such quantum codes can have. While the entanglement-assisted stabilizer formalism overcomes this limitation by exploiting maximally entangled states (ebits), excessive reliance on ebits is a substantial obstacle to implementation. This paper gives necessary and sufficient conditions for the existence of quantum LDPC codes which are obtainable from pairs of identical LDPC codes and consume only one ebit, and studies the spectrum of attainable code parameters.

We develop the theory of entanglement-assisted quantum error correcting codes (EAQECCs), a generalization of the stabilizer formalism to the setting in which the sender and receiver have access to pre-shared entanglement. Conventional stabilizer codes are equivalent to self-orthogonal symplectic codes. In contrast, EAQECCs do not require self-orthogonality, which greatly simplifies their construction. We show how any classical quaternary block code can be made into a EAQECC. Furthermore, the error-correcting power of the quantum codes follows directly from the power of the classical codes.

Physical Review A, 2013

If entanglement is available, the error-correcting ability of quantum codes can be increased. We show how to optimize the minimum distance of an entanglement-assisted quantum error-correcting (EAQEC) code, obtained by adding ebits to a standard quantum error-correcting code, over different encoding operators. By this encoding optimization procedure, we found several new EAQEC codes, including a family of [[n, 1, n; n − 1]] EAQEC codes for n odd and code parameters [[7, 1, 5; 2]], [[7, 1, 5; 3]], [[9, 1, 7; 4]], [[9, 1, 7; 5]], which saturate the quantum singleton bound for EAQEC codes.

2010

This paper develops a general method for constructing entanglement-assisted quantum low-density parity-check (LDPC) codes, which is based on combinatorial design theory. Explicit constructions are given for entanglement-assisted quantum error-correcting codes (EAQECCs) with many desirable properties. These properties include the requirement of only one initial entanglement bit, high error correction performance, high rates, and low decoding complexity. The proposed method produces several infinite families of new codes with a wide variety of parameters and entanglement requirements. Our framework encompasses the previously known entanglement-assisted quantum LDPC codes having the best error correction performance and many new codes with better block error rates in simulations over the depolarizing channel. We also determine important parameters of several well-known classes of quantum and classical LDPC codes for previously unsettled cases.

Physical Review A, 2009

Via explicit examples we show that the pre-existing entanglement can really enhance ͑not only behave as an assistance for͒ the efficiency of the quantum error-correcting codes ͑QECCs͒ in a single block of encoding or decoding as well as help in beating the quantum Hamming bound. A systematic approach to constructing entanglement-assisted ͑or enhanced͒ quantum error-correcting codes ͑EAQECCs͒ via graph states is also presented, and an infinite family of entanglement-enhanced codes has been constructed. Furthermore we generalize the EAQECCs to the case of not-so-perfectly protected qubit and introduce the quantity infidelity as a figure of merit and show that the EAQECCs also outperform the ordinary QECCs.

Physical Review A, 2013

Errors are inevitable during all kinds quantum informational tasks and quantum error-correcting codes (QECCs) are powerful tools to fight various quantum noises. For standard QECCs physical systems have the same number of energy levels. Here we shall propose QECCs over mixed alphabets, i.e., physical systems of different dimensions, and investigate their constructions as well as their quantum Singleton bound. We propose two kinds of constructions: a graphical construction based a graph-theoretical object composite coding clique and a projection-based construction. We illustrate our ideas using two alphabets by finding out some 1-error correcting or detecting codes over mixed alphabets, e.g., optimal ((6, 8, 3)) 4 5 2 1 , ((6, 4, 3)) 4 4 2 2 and ((5, 16, 2)) 4 3 2 2 code and suboptimal ((5, 9, 2)) 3 4 2 1 code. Our methods also shed light to the constructions of standard QECCs, e.g., the construction of the optimal ((6, 16, 3))4 code as well as the optimal ((2n + 3, p 2n+1 , 2))p codes with p = 4k.