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A090899
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Number of nonisomorphic indecomposable self-dual quantum codes on n qubits.
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7
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1, 1, 1, 2, 4, 11, 26, 101, 440, 3132, 40457, 1274068
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OFFSET
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1,4
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COMMENTS
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Also number of nonisomorphic indecomposable self-dual codes of Type 4^H+ and length n.
Each self-dual (additive) quantum code of length n stabilizes an essentially unique quantum state on n qubits, the 2^n coefficients of which can be assumed to take values in {0,1,-1}. It also corresponds to a "quantum" set of n lines in PG(n-1,2): the Grassmannian coordinates of these lines sum to zero. A related sequence is the number of nonisomorphic (possibly decomposable) self-dual quantum codes on n qubits, A094927.
Also the number of equivalence classes of connected graphs on n nodes up to sequences of local complement ation (or vertex neighborhood complementation) and isomorphism.
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REFERENCES
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A. Bouchet, Graphic presentations of isotropic systems, J. Combin. Theory, Ser. B, 45, (1988), 58-76.
Lars Eirik Danielsen and Matthew G. Parker, Spectral Orbits and Peak-to-Average Power Ratio of Boolean Functions with respect to the {I,H,N}^n Transform (2005), http://arxiv.org/abs/cs/0504102. In Sequences and Their Applications-SETA 2004, Lecture Notes in Computer Science, Volume 3486/2005, Springer-Verlag. [Added by N. J. A. Sloane, Jul 08 2009]
L. E. Danielsen and M. G. Parker, On the classification of all self-dual additive codes over GF(4) of length up to 12, Preprint 2005.
David G. Glynn and Johannes G. Maks, The classification of self-dual quantum codes of length <= 9, preprint.
D. M. Schlingemann, Stabilizer codes can be represented as graph codes, Quant. Inf. Comp. 2, 307.
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LINKS
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Table of n, a(n) for n=1..12.
A. R. Calderbank, E. M. Rains, P. W. Shor and N. J. A. Sloane, Quantum Error Correction Via Codes Over GF(4), IEEE Trans. Inform. Theory, 44 (1998), 1369-1387.
Lars Eirik Danielsen, Database of Self-Dual Quantum Codes.
L. E. Danielsen, T. A. Gulliver, M. G. Parker, Aperiodic Propagation Criteria for Boolean Functions, preprint, 2004.
David G. Glynn and Johannes G. Maks, Quantum Error Correction Project (Aotearoa), ClassSD3.pdf.
M. Hein, J. Eisert and H. J. Briegel. Multi-party entanglement in graph states, Phys. Rev. A (3) 69 (2004), no. 6, 062311, 20 pp.
G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
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EXAMPLE
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For four qubits there are two nonisomorphic self-dual quantum codes corresponding to the complete graph and the circuit on four vertices.
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CROSSREFS
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Cf. A094927, A110302, A110306, A151824-A151827.
Sequence in context: A148129 A123432 A151398 * A159338 A159339 A159337
Adjacent sequences: A090896 A090897 A090898 * A090900 A090901 A090902
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KEYWORD
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hard,nonn
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AUTHOR
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David G Glynn (dglynn(AT)mac.com), Feb 26 2004
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EXTENSIONS
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Extended from 9 to 12 terms by Lars Eirik Danielsen (larsed(AT)ii.uib.no) and Matthew G. Parker (matthew(AT)ii.uib.no), Jun 17, 2004.
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STATUS
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approved
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