A094927 -id:A094927 - cp's OEIS frontend

A090899 Number of nonisomorphic indecomposable self-dual quantum codes on n qubits.

1, 1, 1, 2, 4, 11, 26, 101, 440, 3132, 40457, 1274068

Offset: 1

David G Glynn (dglynn(AT)mac.com), Feb 26 2004

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.

			For four qubits there are two nonisomorphic self-dual quantum codes corresponding to the complete graph and the circuit on four vertices.

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.

A. Bouchet, Graphic presentations of isotropic systems, J. Combin. Theory, Ser. B, 45, (1988), 58-76.
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.
L. E. Danielsen and M. G. Parker, On the classification of all self-dual additive codes over GF(4) of length up to 12, Journal of Combinatorial Theory, Series A, Volume 113, Issue 7, October 2006, Pages 1351-1367
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), 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]
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.

Cf. A094927, A110302, A110306, A151824-A151827.

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

A110302 Number of inequivalent indecomposable self-dual codes of Type {4^H+}_II and length 2n.

Original entry on oeis.org

1, 1, 1, 4, 14, 103, 2926

Offset: 0

N. J. A. Sloane, Sep 09 2005

nonn

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.

G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.

Cf. A090899, A094927, A110306.

A110306 Number of inequivalent (indecomposable or decomposable) self-dual codes of Type {4^H+}_II and length 2n.

Original entry on oeis.org

1, 1, 2, 6, 21, 128, 3079

Offset: 0

N. J. A. Sloane, Sep 09 2005

L. E. Danielsen and M. G. Parker, On the classification of all self-dual additive codes over GF(4) of length up to 12, arXiv:math/0504522 [math.CO], 2005-2006.
L. E. Danielsen and M. G. Parker, On the classification of all self-dual additive codes over GF(4) of length up to 12, J. Combin. Theory A 113 (7) (2006) 1351-1367.
W. C. Huffman, On the classification and enumeration of self-dual codes, Finite Fields Applic., 11 (2005), 451-490.
W. C. Huffman, Additive self-dual codes over F_4 with an automorphism of odd prime order, Adv. Math. Commun., 1 (2007), 357-398.
G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.

Cf. A090899, A094927, A110302.

A105687 Number of inequivalent codes attaining highest minimal Hamming distance of any Type 4^H+ Hermitian additive self-dual code over GF(4) of length n.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 4, 5, 8, 120, 1, 1

Offset: 1

N. J. A. Sloane, May 06 2005

nonn

C. Bachoc and P. Gaborit, On extremal additive F_4 codes of length 10 to 18, in International Workshop on Coding and Cryptography (Paris, 2001), Electron. Notes Discrete Math. 6 (2001), 10 pp.
P. Gaborit, W. C. Huffman, J.-L. Kim and V. S. Pless, On additive GF(4) codes, in Codes and Association Schemes (Piscataway, NJ, 1999), A. Barg and S. Litsyn, eds., Amer. Math. Soc., Providence, RI, 2001, pp. 135-149.
G. Hoehn, Self-dual codes over the Kleinian four-group, Math. Ann. 327 (2003), 227-255.

G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
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.
L. E. Danielsen, Database of Self-Dual Quantum Codes.
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.
E. M. Rains and N. J. A. Sloane, Self-dual codes, pp. 177-294 of Handbook of Coding Theory, Elsevier, 1998 (Abstract, pdf, ps).

Cf. A094927, A090899, A105674, A105675, A105676, A105677, A105678, A016729, A066016, A105681, A105682.
A016729 gives the minimal distance of these codes.
A094927 gives the number of inequivalent codes of any distance.

Corrected and extended to 12 terms by Lars Eirik Danielsen (larsed(AT)ii.uib.no) and Matthew G. Parker (matthew(AT)ii.uib.no), Jun 30 2005

cp's OEIS Frontend

A090899 Number of nonisomorphic indecomposable self-dual quantum codes on n qubits.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Extensions

A110302 Number of inequivalent indecomposable self-dual codes of Type {4^H+}_II and length 2n.

Views

Author

Keywords

References

Links

Crossrefs

A110306 Number of inequivalent (indecomposable or decomposable) self-dual codes of Type {4^H+}_II and length 2n.

Views

Author

Keywords

Links

Crossrefs

A105687 Number of inequivalent codes attaining highest minimal Hamming distance of any Type 4^H+ Hermitian additive self-dual code over GF(4) of length n.

Views

Author

Keywords

References

Links

Crossrefs

Extensions