Abstract
The combinatorial aspects of quantum codes were demonstrated in the study of decay processes of certain quantum systems used in the newly emerging field of quantum computing. Among them, the configuration of t-spontaneous emission error design (t-SEED) was proposed to correct errors caused by quantum jumps. The number of designs (dimension) in a t-SEED corresponds to the number of orthogonal basis states in a quantum jump code. In this paper the upper and lower bounds on the dimensions of 3-\((v,\,4;\,m)\) SEEDs are studied and the necessary and sufficient conditions for 3-SEEDs attaining the upper bounds are described. Several new combinatorial constructions are presented for general t-SEEDs and lots of t-SEEDs of new parameters with \(t\ge 3\) are shown to exist.
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Acknowledgements
The authors would like to thank the anonymous referees for many valuable comments. Supported by NSFC Grants 11431003 and 11571034.
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Communicated by C. Mitchell.
Appendix
Appendix
Four block sets in Lemma 2.14.
\(0\ 1\ 2\ 3\) | \(0\ 1\ 5\ 6\) | \(0\ 1\ 7\ 8\) | \(0\ 1\ 9\ 10\) | \(0\ 1\ 11\ 13\) | \(0\ 1\ 12\ 14\) | \(0\ 2\ 4\ 5\) |
\(0\ 2\ 6\ 12\) | \(0\ 2\ 7\ 11\) | \(0\ 2\ 9\ 13\) | \(0\ 2\ 10\ 14\) | \(0\ 3\ 4\ 10\) | \(0\ 3\ 5\ 7\) | \(0\ 3\ 6\ 8\) |
\(0\ 3\ 9\ 11\) | \(0\ 3\ 12\ 13\) | \(0\ 4\ 6\ 9\) | \(0\ 4\ 7\ 12\) | \(0\ 4\ 8\ 13\) | \(0\ 4\ 11\ 14\) | \(0\ 5\ 8\ 11\) |
\(0\ 5\ 9\ 12\) | \(0\ 5\ 13\ 14\) | \(0\ 6\ 7\ 14\) | \(0\ 6\ 10\ 11\) | \(0\ 7\ 10\ 13\) | \(0\ 8\ 9\ 14\) | \(0\ 8\ 10\ 12\) |
\(1\ 2\ 4\ 9\) | \(1\ 2\ 6\ 13\) | \(1\ 2\ 7\ 10\) | \(1\ 2\ 8\ 12\) | \(1\ 2\ 11\ 14\) | \(1\ 3\ 4\ 14\) | \(1\ 3\ 5\ 8\) |
\(1\ 3\ 6\ 7\) | \(1\ 3\ 10\ 13\) | \(1\ 3\ 11\ 12\) | \(1\ 4\ 5\ 13\) | \(1\ 4\ 6\ 8\) | \(1\ 4\ 7\ 11\) | \(1\ 4\ 10\ 12\) |
\(1\ 5\ 7\ 12\) | \(1\ 5\ 9\ 14\) | \(1\ 5\ 10\ 11\) | \(1\ 6\ 9\ 12\) | \(1\ 6\ 10\ 14\) | \(1\ 7\ 9\ 13\) | \(1\ 8\ 9\ 11\) |
\(1\ 8\ 13\ 14\) | \(2\ 3\ 4\ 13\) | \(2\ 3\ 5\ 12\) | \(2\ 3\ 7\ 14\) | \(2\ 3\ 8\ 9\) | \(2\ 3\ 10\ 11\) | \(2\ 4\ 6\ 7\) |
\(2\ 4\ 8\ 11\) | \(2\ 4\ 12\ 14\) | \(2\ 5\ 6\ 11\) | \(2\ 5\ 7\ 9\) | \(2\ 5\ 8\ 14\) | \(2\ 5\ 10\ 13\) | \(2\ 6\ 8\ 10\) |
\(2\ 6\ 9\ 14\) | \(2\ 7\ 8\ 13\) | \(2\ 9\ 10\ 12\) | \(2\ 11\ 12\ 13\) | \(3\ 4\ 5\ 9\) | \(3\ 4\ 6\ 11\) | \(3\ 4\ 8\ 12\) |
\(3\ 5\ 6\ 13\) | \(3\ 5\ 10\ 14\) | \(3\ 6\ 9\ 10\) | \(3\ 6\ 12\ 14\) | \(3\ 7\ 8\ 10\) | \(3\ 7\ 9\ 12\) | \(3\ 7\ 11\ 13\) |
\(3\ 8\ 11\ 14\) | \(3\ 9\ 13\ 14\) | \(4\ 5\ 6\ 14\) | \(4\ 5\ 7\ 10\) | \(4\ 5\ 11\ 12\) | \(4\ 6\ 10\ 13\) | \(4\ 7\ 8\ 9\) |
\(4\ 7\ 13\ 14\) | \(4\ 8\ 10\ 14\) | \(4\ 9\ 10\ 11\) | \(4\ 9\ 12\ 13\) | \(5\ 6\ 7\ 8\) | \(5\ 6\ 10\ 12\) | \(5\ 7\ 11\ 14\) |
\(5\ 8\ 9\ 10\) | \(5\ 8\ 12\ 13\) | \(5\ 9\ 11\ 13\) | \(6\ 7\ 9\ 11\) | \(6\ 7\ 12\ 13\) | \(6\ 8\ 9\ 13\) | \(6\ 8\ 11\ 12\) |
\(6\ 11\ 13\ 14\) | \(7\ 8\ 12\ 14\) | \(7\ 9\ 10\ 14\) | \(7\ 10\ 11\ 12\) | \(8\ 10\ 11\ 13\) | \(9\ 11\ 12\ 14\) | \(10\ 12\ 13\ 14\) |
\(0\ 1\ 2\ 9\) | \(0\ 1\ 3\ 6\) | \(0\ 1\ 5\ 13\) | \(0\ 1\ 7\ 12\) | \(0\ 1\ 8\ 11\) | \(0\ 1\ 10\ 14\) | \(0\ 2\ 3\ 4\) |
\(0\ 2\ 5\ 12\) | \(0\ 2\ 6\ 14\) | \(0\ 2\ 7\ 10\) | \(0\ 2\ 11\ 13\) | \(0\ 3\ 5\ 8\) | \(0\ 3\ 7\ 11\) | \(0\ 3\ 9\ 12\) |
\(0\ 3\ 10\ 13\) | \(0\ 4\ 5\ 9\) | \(0\ 4\ 6\ 8\) | \(0\ 4\ 7\ 13\) | \(0\ 4\ 10\ 11\) | \(0\ 4\ 12\ 14\) | \(0\ 5\ 6\ 7\) |
\(0\ 5\ 11\ 14\) | \(0\ 6\ 9\ 11\) | \(0\ 6\ 10\ 12\) | \(0\ 7\ 8\ 14\) | \(0\ 8\ 9\ 10\) | \(0\ 8\ 12\ 13\) | \(0\ 9\ 13\ 14\) |
\(1\ 2\ 3\ 11\) | \(1\ 2\ 4\ 6\) | \(1\ 2\ 7\ 13\) | \(1\ 2\ 8\ 14\) | \(1\ 2\ 10\ 12\) | \(1\ 3\ 4\ 12\) | \(1\ 3\ 5\ 10\) |
\(1\ 3\ 7\ 8\) | \(1\ 3\ 13\ 14\) | \(1\ 4\ 5\ 7\) | \(1\ 4\ 8\ 13\) | \(1\ 4\ 9\ 10\) | \(1\ 4\ 11\ 14\) | \(1\ 5\ 6\ 14\) |
\(1\ 5\ 8\ 9\) | \(1\ 5\ 11\ 12\) | \(1\ 6\ 7\ 9\) | \(1\ 6\ 8\ 12\) | \(1\ 6\ 10\ 13\) | \(1\ 7\ 10\ 11\) | \(1\ 9\ 11\ 13\) |
\(1\ 9\ 12\ 14\) | \(2\ 3\ 5\ 7\) | \(2\ 3\ 8\ 12\) | \(2\ 3\ 9\ 13\) | \(2\ 3\ 10\ 14\) | \(2\ 4\ 5\ 14\) | \(2\ 4\ 7\ 11\) |
\(2\ 4\ 8\ 9\) | \(2\ 4\ 12\ 13\) | \(2\ 5\ 6\ 13\) | \(2\ 5\ 8\ 11\) | \(2\ 5\ 9\ 10\) | \(2\ 6\ 7\ 8\) | \(2\ 6\ 9\ 12\) |
\(2\ 6\ 10\ 11\) | \(2\ 7\ 9\ 14\) | \(2\ 8\ 10\ 13\) | \(2\ 11\ 12\ 14\) | \(3\ 4\ 5\ 13\) | \(3\ 4\ 6\ 10\) | \(3\ 4\ 8\ 14\) |
\(3\ 4\ 9\ 11\) | \(3\ 5\ 6\ 12\) | \(3\ 5\ 9\ 14\) | \(3\ 6\ 7\ 13\) | \(3\ 6\ 8\ 9\) | \(3\ 6\ 11\ 14\) | \(3\ 7\ 9\ 10\) |
\(3\ 7\ 12\ 14\) | \(3\ 8\ 10\ 11\) | \(3\ 11\ 12\ 13\) | \(4\ 5\ 6\ 11\) | \(4\ 5\ 10\ 12\) | \(4\ 6\ 7\ 14\) | \(4\ 6\ 9\ 13\) |
\(4\ 7\ 8\ 10\) | \(4\ 7\ 9\ 12\) | \(4\ 8\ 11\ 12\) | \(4\ 10\ 13\ 14\) | \(5\ 6\ 8\ 10\) | \(5\ 7\ 8\ 12\) | \(5\ 7\ 9\ 11\) |
\(5\ 7\ 10\ 14\) | \(5\ 8\ 13\ 14\) | \(5\ 9\ 12\ 13\) | \(5\ 10\ 11\ 13\) | \(6\ 7\ 11\ 12\) | \(6\ 8\ 11\ 13\) | \(6\ 9\ 10\ 14\) |
\(6\ 12\ 13\ 14\) | \(7\ 8\ 9\ 13\) | \(7\ 10\ 12\ 13\) | \(7\ 11\ 13\ 14\) | \(8\ 9\ 11\ 14\) | \(8\ 10\ 12\ 14\) | \(9\ 10\ 11\ 12\) |
\(0\ 1\ 2\ 10\) | \(0\ 1\ 3\ 12\) | \(0\ 1\ 5\ 14\) | \(0\ 1\ 6\ 8\) | \(0\ 1\ 7\ 11\) | \(0\ 1\ 9\ 13\) | \(0\ 2\ 3\ 5\) |
\(0\ 2\ 4\ 12\) | \(0\ 2\ 6\ 11\) | \(0\ 2\ 7\ 13\) | \(0\ 2\ 9\ 14\) | \(0\ 3\ 4\ 13\) | \(0\ 3\ 6\ 7\) | \(0\ 3\ 8\ 9\) |
\(0\ 3\ 10\ 11\) | \(0\ 4\ 5\ 6\) | \(0\ 4\ 7\ 14\) | \(0\ 4\ 8\ 11\) | \(0\ 4\ 9\ 10\) | \(0\ 5\ 7\ 8\) | \(0\ 5\ 9\ 11\) |
\(0\ 5\ 12\ 13\) | \(0\ 6\ 9\ 12\) | \(0\ 6\ 10\ 14\) | \(0\ 7\ 10\ 12\) | \(0\ 8\ 10\ 13\) | \(0\ 8\ 12\ 14\) | \(0\ 11\ 13\ 14\) |
\(1\ 2\ 3\ 14\) | \(1\ 2\ 4\ 11\) | \(1\ 2\ 6\ 7\) | \(1\ 2\ 8\ 13\) | \(1\ 2\ 9\ 12\) | \(1\ 3\ 4\ 10\) | \(1\ 3\ 5\ 6\) |
\(1\ 3\ 7\ 13\) | \(1\ 3\ 8\ 11\) | \(1\ 4\ 5\ 9\) | \(1\ 4\ 6\ 13\) | \(1\ 4\ 7\ 12\) | \(1\ 4\ 8\ 14\) | \(1\ 5\ 7\ 10\) |
\(1\ 5\ 8\ 12\) | \(1\ 5\ 11\ 13\) | \(1\ 6\ 9\ 10\) | \(1\ 6\ 12\ 14\) | \(1\ 7\ 8\ 9\) | \(1\ 9\ 11\ 14\) | \(1\ 10\ 11\ 12\) |
\(1\ 10\ 13\ 14\) | \(2\ 3\ 4\ 8\) | \(2\ 3\ 7\ 11\) | \(2\ 3\ 9\ 10\) | \(2\ 3\ 12\ 13\) | \(2\ 4\ 5\ 13\) | \(2\ 4\ 6\ 14\) |
\(2\ 4\ 7\ 9\) | \(2\ 5\ 6\ 10\) | \(2\ 5\ 7\ 14\) | \(2\ 5\ 8\ 9\) | \(2\ 5\ 11\ 12\) | \(2\ 6\ 8\ 12\) | \(2\ 6\ 9\ 13\) |
\(2\ 7\ 8\ 10\) | \(2\ 8\ 11\ 14\) | \(2\ 10\ 11\ 13\) | \(2\ 10\ 12\ 14\) | \(3\ 4\ 5\ 12\) | \(3\ 4\ 6\ 9\) | \(3\ 4\ 11\ 14\) |
\(3\ 5\ 7\ 9\) | \(3\ 5\ 8\ 14\) | \(3\ 5\ 10\ 13\) | \(3\ 6\ 8\ 10\) | \(3\ 6\ 11\ 12\) | \(3\ 6\ 13\ 14\) | \(3\ 7\ 8\ 12\) |
\(3\ 7\ 10\ 14\) | \(3\ 9\ 11\ 13\) | \(3\ 9\ 12\ 14\) | \(4\ 5\ 7\ 11\) | \(4\ 5\ 10\ 14\) | \(4\ 6\ 7\ 8\) | \(4\ 6\ 10\ 11\) |
\(4\ 7\ 10\ 13\) | \(4\ 8\ 9\ 13\) | \(4\ 8\ 10\ 12\) | \(4\ 9\ 11\ 12\) | \(4\ 12\ 13\ 14\) | \(5\ 6\ 7\ 12\) | \(5\ 6\ 8\ 13\) |
\(5\ 6\ 11\ 14\) | \(5\ 8\ 10\ 11\) | \(5\ 9\ 10\ 12\) | \(5\ 9\ 13\ 14\) | \(6\ 7\ 9\ 14\) | \(6\ 7\ 11\ 13\) | \(6\ 8\ 9\ 11\) |
\(6\ 10\ 12\ 13\) | \(7\ 8\ 13\ 14\) | \(7\ 9\ 10\ 11\) | \(7\ 9\ 12\ 13\) | \(7\ 11\ 12\ 14\) | \(8\ 9\ 10\ 14\) | \(8\ 11\ 12\ 13\) |
\(0\ 1\ 2\ 11\) | \(0\ 1\ 3\ 7\) | \(0\ 1\ 5\ 9\) | \(0\ 1\ 6\ 14\) | \(0\ 1\ 8\ 13\) | \(0\ 1\ 10\ 12\) | \(0\ 2\ 3\ 13\) |
\(0\ 2\ 4\ 9\) | \(0\ 2\ 5\ 7\) | \(0\ 2\ 6\ 10\) | \(0\ 2\ 12\ 14\) | \(0\ 3\ 4\ 12\) | \(0\ 3\ 5\ 6\) | \(0\ 3\ 8\ 11\) |
\(0\ 3\ 9\ 10\) | \(0\ 4\ 5\ 14\) | \(0\ 4\ 6\ 11\) | \(0\ 4\ 7\ 8\) | \(0\ 4\ 10\ 13\) | \(0\ 5\ 8\ 12\) | \(0\ 5\ 11\ 13\) |
\(0\ 6\ 7\ 12\) | \(0\ 6\ 8\ 9\) | \(0\ 7\ 10\ 11\) | \(0\ 7\ 13\ 14\) | \(0\ 8\ 10\ 14\) | \(0\ 9\ 11\ 14\) | \(0\ 9\ 12\ 13\) |
\(1\ 2\ 3\ 8\) | \(1\ 2\ 4\ 7\) | \(1\ 2\ 6\ 12\) | \(1\ 2\ 9\ 13\) | \(1\ 2\ 10\ 14\) | \(1\ 3\ 4\ 13\) | \(1\ 3\ 5\ 12\) |
\(1\ 3\ 6\ 10\) | \(1\ 3\ 11\ 14\) | \(1\ 4\ 5\ 10\) | \(1\ 4\ 6\ 9\) | \(1\ 4\ 8\ 11\) | \(1\ 4\ 12\ 14\) | \(1\ 5\ 6\ 8\) |
\(1\ 5\ 7\ 11\) | \(1\ 5\ 13\ 14\) | \(1\ 6\ 7\ 13\) | \(1\ 7\ 8\ 12\) | \(1\ 7\ 9\ 10\) | \(1\ 8\ 9\ 14\) | \(1\ 9\ 11\ 12\) |
\(1\ 10\ 11\ 13\) | \(2\ 3\ 4\ 5\) | \(2\ 3\ 7\ 10\) | \(2\ 3\ 9\ 14\) | \(2\ 3\ 11\ 12\) | \(2\ 4\ 6\ 13\) | \(2\ 4\ 8\ 12\) |
\(2\ 4\ 11\ 14\) | \(2\ 5\ 6\ 14\) | \(2\ 5\ 8\ 13\) | \(2\ 5\ 9\ 12\) | \(2\ 5\ 10\ 11\) | \(2\ 6\ 7\ 9\) | \(2\ 6\ 8\ 11\) |
\(2\ 7\ 8\ 14\) | \(2\ 7\ 11\ 13\) | \(2\ 8\ 9\ 10\) | \(2\ 10\ 12\ 13\) | \(3\ 4\ 6\ 14\) | \(3\ 4\ 8\ 9\) | \(3\ 4\ 10\ 11\) |
\(3\ 5\ 7\ 14\) | \(3\ 5\ 8\ 10\) | \(3\ 5\ 9\ 13\) | \(3\ 6\ 7\ 8\) | \(3\ 6\ 9\ 12\) | \(3\ 6\ 11\ 13\) | \(3\ 7\ 9\ 11\) |
\(3\ 7\ 12\ 13\) | \(3\ 8\ 12\ 14\) | \(3\ 10\ 13\ 14\) | \(4\ 5\ 6\ 7\) | \(4\ 5\ 9\ 11\) | \(4\ 5\ 12\ 13\) | \(4\ 6\ 8\ 10\) |
\(4\ 7\ 9\ 13\) | \(4\ 7\ 10\ 14\) | \(4\ 7\ 11\ 12\) | \(4\ 8\ 13\ 14\) | \(4\ 9\ 10\ 12\) | \(5\ 6\ 10\ 13\) | \(5\ 6\ 11\ 12\) |
\(5\ 7\ 8\ 9\) | \(5\ 7\ 10\ 12\) | \(5\ 8\ 11\ 14\) | \(5\ 9\ 10\ 14\) | \(6\ 7\ 11\ 14\) | \(6\ 8\ 12\ 13\) | \(6\ 9\ 10\ 11\) |
\(6\ 9\ 13\ 14\) | \(6\ 10\ 12\ 14\) | \(7\ 8\ 10\ 13\) | \(7\ 9\ 12\ 14\) | \(8\ 9\ 11\ 13\) | \(8\ 10\ 11\ 12\) | \(11\ 12\ 13\ 14\) |
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Zhou, J., Chang, Y. Bounds and constructions of t-spontaneous emission error designs. Des. Codes Cryptogr. 85, 249–271 (2017). https://doi.org/10.1007/s10623-016-0300-x
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DOI: https://doi.org/10.1007/s10623-016-0300-x