基于分圆陪集的量子BCH码的构造

doi:10.11959/j.issn.1000-436x.2021194

Abstract

Abstract:

Quantum-error-correcting code can overcome quantum decoherence efficiently, which is the key technology to realize quantum computers.A series of quantum BCH code was proposed based on classical codes.First, a general way of well-chosen cyclotomic coset was introduced.A sufficient condition was given to calculate the number of elements in cyclotomic coset.Then, a series of quantum BCH (Bose-Chaudhuri-Hocquenghem) code over finite field F_q was constructed by CSS (Calderbank-Shor-Steane) construction and Steane construction.The results were extended to finite field F_q² with Hermitian construction.Compared with the results in literature,the range of introduced cyclotomic coset is more wide, and the new quantum BCH code has higher dimensions and better lower bounds on minimum distances.Furthermore, a family of quantum maximum distance separable (quantum MDS) code over any finite fields is constructed.

Key words: quantum BCH code, cyclotomic coset, Steane construction, Hermitian construction

CLC Number:

TN929.12

Lijuan XING, Zhuo LI. Construction of quantum BCH code based on cyclotomic coset[J]. Journal on Communications, 2021, 42(10): 182-188.

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References 14

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n	r	c	t	新的量子码
56	4	3	3	[[56,41,d≥5]]₁₃
42	3	2	2	[[42,31,d≥4]]₁₃
84	6	5	5	[[84,62,d≥7]]₁₃

构造的量子码	文献[10]	文献[11]
[[40,32,d≥4]]₉	[[40,28,d≥4]]₉	[[40,30,d≥4]]₉
[[40,36,3]]₉(MDS)	—	—
[[60,48,d≥5]]₁₁	[[60,44,d≥5]]₁₁	[[60,46,d≥5]]₁₁
[[60,56,3]]₁₁(MDS)	—	—
[[84,68,d≥6]]₁₃	[[84,60,d≥6]]₁₃	[[84,62,d≥6]]₁₃
[[84,80,3]]₁₃(MDS)	—	—

构造的量子码	文献[1]
[[40,20,d≥6]]₃	[[40,26,d≥5]]₃
[[312,262,d≥14]]₅	[[312,266,d≥13]]₅
[[1200,1104,d≥26]]₅	[[1200,1106,d≥25]]₇

构造的量子码	文献[2]	文献[14]
[[40,34,d≥3]]₃	[[40,34,d≥3]]₃	[[41,33,d≥3]]₃
[[312,306,d≥3]]₅	[[312,306,d≥3]]₅	[[313,305,d≥3]]₅
[[312,298,d≥5]]₅	[[312,298,d≥5]]₅	[[313,297,d≥5]]₅
[[1200,1194,d≥3]]₇	[[1200,1194,d≥3]]₇	[[1201,1193,d≥3]]₇
[[1200,1186,d≥5]]₇	[[1200,1186,d≥5]]₇	[[1201,1185,d≥3]]₇
[[1200,1178,d≥7]]₇	[[1200,1178,d≥7]]₇	[[1201,1177,d≥3]]₇

Construction of quantum BCH code based on cyclotomic coset

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