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The minimum distance of d=4 quantum MDS codes

Author: GuoYanJu
Tutor: TangYuanSheng; LuoJinQuan
School: Yangzhou University
Course: Applied Mathematics
Keywords: quantum error-correcting codes quantum MDS codes Hermitian self-orthogonal codes
CLC: O413
Type: Master's thesis
Year: 2011
Downloads: 14
Quote: 0
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Abstract


After the emergence of quantum computers, there are some difficulties in encoding in the quantum environment because of the quantum no-cloning feature. Shor and Steane avoided these problems by means of some ingenious measures and proposed quantum error-correcting codes. Quantum error correcting codes are closely related to secure communication, it is an important measure to improve the reliability of transmitting the information, thus the quantum error correction in quantum information theory occupies a very important position。Like Classical coding theory, how to find out better parameters of quantum codes has become one central study.Quantum MDS code is a special code satisfying the quantum Singleton bound. It has a very strong error correction capability, especially when the code length is not very long, its performance is very close to the theoretical value. In addition, it contained a good algebraic structure, can be constructed conveniently, easy to encode and decode, and has great practicability. So in recent years Quantum MDS code is applied into many communications systems.At present, there are two main methods to construct the quantum MDS codes. One method is to construct by the known classical codes, we can first construct quantum stabilizer codes, algebraic geometric codes, classical self-orthogonal codes, or symmetry, Euclidean Reed, MDS Hermitian self-orthogonal codes, generalized RS codes, then educe the quantum MDS codes; The other method constructing quantum codes is to make use of the graph theory, although the constructing process seems difficulty, it is very meaningful.Literature [4] has proved that when the yard meet certain condition all the q element of quantum MDS codes for the minimum distance d=3exist, and when d=4the existence of quantum MDS codes remains to be verified, if exists, we must construct a generator matrix, and started this discussion in the article. In this paper, by combination with the literature [4] Theorem and conclusions, together with the construction method in the literature [3] to construct some classical Hermitian self-orthogonal code in a finite field, by which to get some quantum MDS codes. Finite fields is given in the article on the Hermitian self-orthogonal codes and finite fields of quantum MDS codes corresponding relationship between them; Then we give some minimum distance d=4quantum MDS code generator matrix by using mathematical software Magma, and list all the failure cases when considering all the circumstances.

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CLC: > Mathematical sciences and chemical > Physics > Theoretical Physics > Quantum theory
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