All the Stabilizer Codes of Distance 3

We give necessary and sufficient conditions for the existence of stabilizer codes [[n, k, 3]] of distance 3 for qubits: n-k ≥ ⌈log 2 (3n + 1)⌉ + ∈ n , where ∈ n = 1 if n = 84 m -1/3 + {±1, 2} or n = 4 m+2 -1/3 -{1, 2, 3} for some integer m ≥ 1 and ∈ n = 0 otherwise. Or equivalently, a code [[n, n -...

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Bibliographic Details
Published in:IEEE transactions on information theory 2013-08, Vol.59 (8), p.5179-5185
Main Authors: Yu, Sixia, Bierbrauer, Jurgen, Ying Dong, Qing Chen, Oh, C. H.
Format: Article
Language:English
Subjects:
1-error correcting stabilizer codes
Applied sciences
Classical and quantum physics: mechanics and fields
Codes
Coding, codes
Educational institutions
Error correction codes
Exact sciences and technology
Frequency modulation
Generators
Indexes
Information theory
Information, signal and communications theory
optimal codes
Physics
quantum error correction
quantum Hamming bound
Quantum information
Signal and communications theory
Telecommunications and information theory
Vectors
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Summary:We give necessary and sufficient conditions for the existence of stabilizer codes [[n, k, 3]] of distance 3 for qubits: n-k ≥ ⌈log 2 (3n + 1)⌉ + ∈ n , where ∈ n = 1 if n = 84 m -1/3 + {±1, 2} or n = 4 m+2 -1/3 -{1, 2, 3} for some integer m ≥ 1 and ∈ n = 0 otherwise. Or equivalently, a code [[n, n - r, 3]] exists if and only if n ≤ (4 r - 1)/3, (4 r - 1)/3 - n ∉ {1, 2, 3} for even r and n ≤ 8 (4 r-3 -1)/3, 8(4 -3 - 1)/3 - n ≠ 1 for odd r. Given an arbitrary length n, we present an explicit construction for an optimal quantum stabilizer code of distance 3 that saturates the above bound.
ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2013.2259138