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%I #20 Jan 03 2020 14:21:28
%S 1,1,1,2,4,11,26,101,440,3132,40457,1274068
%N Number of nonisomorphic indecomposable self-dual quantum codes on n qubits.
%C Also number of nonisomorphic indecomposable self-dual codes of Type 4^H+ and length n.
%C 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.
%C 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.
%D David G. Glynn and Johannes G. Maks, The classification of self-dual quantum codes of length <= 9, preprint.
%D D. M. Schlingemann, Stabilizer codes can be represented as graph codes, Quant. Inf. Comp. 2, 307.
%H A. Bouchet, <a href="http://dx.doi.org/10.1016/0095-8956(88)90055-X">Graphic presentations of isotropic systems</a>, J. Combin. Theory, Ser. B, 45, (1988), 58-76.
%H A. R. Calderbank, E. M. Rains, P. W. Shor and N. J. A. Sloane, <a href="https://arxiv.org/abs/quant-ph/9608006">Quantum Error Correction Via Codes Over GF(4)</a>, IEEE Trans. Inform. Theory, 44 (1998), 1369-1387.
%H Lars Eirik Danielsen, <a href="http://www.ii.uib.no/~larsed/vncorbits/">Database of Self-Dual Quantum Codes</a>.
%H L. E. Danielsen, T. A. Gulliver, M. G. Parker, <a href="http://www.ii.uib.no/~matthew/GenDiff2.pdf">Aperiodic Propagation Criteria for Boolean Functions</a>, preprint, 2004.
%H L. E. Danielsen and M. G. Parker, <a href="http://dx.doi.org/10.1016/j.jcta.2005.12.004">On the classification of all self-dual additive codes over GF(4) of length up to 12</a>, Journal of Combinatorial Theory, Series A, Volume 113, Issue 7, October 2006, Pages 1351-1367
%H Lars Eirik Danielsen and Matthew G. Parker, <a href="http://arxiv.org/abs/cs/0504102">Spectral Orbits and Peak-to-Average Power Ratio of Boolean Functions with respect to the {I,H,N}^n Transform</a>, (2005), arxiv: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]
%H David G. Glynn and Johannes G. Maks, <a href="http://homepage.mac.com/dglynn/">Quantum Error Correction Project (Aotearoa)</a>, ClassSD3.pdf.
%H M. Hein, J. Eisert and H. J. Briegel. <a href="http://arXiv.org/quant-ph/0307130">Multi-party entanglement in graph states</a>, Phys. Rev. A (3) 69 (2004), no. 6, 062311, 20 pp.
%H G. Nebe, E. M. Rains and N. J. A. Sloane, <a href="http://neilsloane.com/doc/cliff2.html">Self-Dual Codes and Invariant Theory</a>, Springer, Berlin, 2006.
%e For four qubits there are two nonisomorphic self-dual quantum codes corresponding to the complete graph and the circuit on four vertices.
%Y Cf. A094927, A110302, A110306, A151824-A151827.
%K hard,nonn
%O 1,4
%A David G Glynn (dglynn(AT)mac.com), Feb 26 2004
%E a(10)-a(12) from Lars Eirik Danielsen (larsed(AT)ii.uib.no) and Matthew G. Parker (matthew(AT)ii.uib.no), Jun 17 2004
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