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Introduction

We have written a paper that explains these results [14]. Also see my Master's thesis [1].

Zero-dimensional [[n,0,d]] quantum stabilizer codes corresponds to additive (n,2n,d) codes over GF(4) which are self-dual with respect to a trace inner product [2]. Zero-dimensional quantum codes represent highly entangled single quantum states.

Every self-dual additive code over GF(4) can be represented by a simple undirected graph [3,4]. The graph with adjacency matrix Γ represents the code with generator matrix Γ + ωI. Local Complementation, a simple graph operation, generates the orbits of equivalent self-dual additive codes over GF(4). This was first shown in the context of isotropic systems by Bouchet [3], and later by Glynn et al. [5,6], by Hein et al. [7] and by Van den Nest et al. [8].

Using LC, we can generate the orbits of all inequivalent self-dual additive codes over GF(4) of length n. These codes have previously been classified by Calderbank et al. [1] (up to n=5), by Hein et al. [7] (up to n=7), by Höhn [9] (up to n=7) and by Glynn et al. [6] (up to n=9). For higher lengths, only optimal codes have been classified [10,11]. We have generated all LC orbits for n up to 12, and this database contains a representative of each orbit. The members of an LC orbit are counted up to isomorphism. (We use the program nauty to check for graph isomorphism.)

A graph can also be represented by a quadratic Boolean function. For any Boolean function, we can generate the {I,H,N}n-orbit [12]. For quadratic Boolean functions, the {I,H,N}n-orbit is the LC orbit. Nonquadratic Boolean functions can also be interpreted as quantum codes, but they do not correspond to self-dual additive codes over GF(4). See the Database of Nonquadratic Quantum Codes for orbits with respect to {I,H,N}n and other symmetries. For quadratic and nonquadratic Boolean functions we are interested in PARIHN, the Peak-to-Average power Ratio with respect to the {I,H,N}n transform, and APC, the Aperiodic Propagation Criteria [13]. Both PARIHN and APC of a Boolean function are related to the distance of the corresponding quantum code.

(See also David Glynn's web page.)

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Tables

Number of inequivalent indecomposable (in) and possibly decomposable (tn) self-dual quantum codes:

 123456789101112
in1112411261014403,13240,4571,274,068
tn12361126591826753,99045,1441,323,363

These are sequences A090899 and A094927 in the On-Line Encyclopedia of Integer Sequences. A decomposable code corresponds to an unconnected graph. This database only contains the indecomposable codes, corresponding to connected graphs, since the decomposable codes can easily be constructed by combining indecomposable codes of shorter length.

Number of self-dual quantum codes of various distances:

d\n23456789101112
21123922853632,43626,750611,036
3114116957611,200467,513
41581202,506195,455
5163
61

(See Markus Grassl's web page for bounds on the minimum distance for other lengths and dimensions.)

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File formats

The files are available in various formats: nauty's graph6 format, ANF, Magma and Maple adjacency matrix format. The graph with adjacency matrix Γ represents the code with generator matrix Γ + ωI.

The graph6 files can be used as input to all the utilities in the nauty package. Files of the type "g6 with data" consist of records of tab separated values of the following order:

The ANF files contains Boolean functions in algebraic normal form notation. Example: "03,13,23," is the function "x0x3+x1x3+x2x3", which can also be interpreted as the graph with undirected edges {(0,3),(1,3),(2,3)}.

NEW You can also download one big file containing all codes, including decomposable codes, with all data I have available. The records have the following order:

Download File size
thebigselfdualfile.txt.gz 14MB

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Files

All lengths in compressed archives

File format Download File size
graph6 vncorbits_g6.tar.gz 10MB
g6 with data vncorbits_data.tar.gz 15MB
ANF vncorbits_anf.tar.gz 14MB
Magma vncorbits_magma.tar.gz 18MB
Maple vncorbits_maple.tar.gz 38MB

Single files for each length

n=12 (compressed)

File format Download File size
graph6 vncorbits12.g6.gz 9.7MB
g6 with data vncorbits12.data.gz 14MB
ANF vncorbits12.anf.gz 14MB
Magma vncorbits12.magma.gz 18MB
Maple vncorbits12.maple.gz 37MB

n=11

File format Download File size
graph6 vncorbits11.g6 474KB
g6 with data vncorbits11.data 2.3MB
ANF vncorbits11.anf 2.1MB
Magma vncorbits11.magma 3.9MB
Maple vncorbits11.maple 12MB

n=10

File format Download File size
graph6 vncorbits10.g6 31KB
g6 with data vncorbits10.data 158KB
ANF vncorbits10.anf 130KB
Magma vncorbits10.magma 264KB
Maple vncorbits10.maple 797KB

n=9

File format Download File size
graph6 vncorbits9.g6 3.4KB
g6 with data vncorbits9.data 19KB
ANF vncorbits9.anf 15KB
Magma vncorbits9.magma 30KB
Maple vncorbits9.maple 93KB

n=8

File format Download File size
graph6 vncorbits8.g6 707B
g6 with data vncorbits8.data 3.7KB
ANF vncorbits8.anf 2.8KB
Magma vncorbits8.magma 6.2KB
Maple vncorbits8.maple 18KB

n=7

File format Download File size
graph6 vncorbits7.g6 156B
g6 with data vncorbits7.data 860B
ANF vncorbits7.anf 590B
Magma vncorbits7.magma 1.4KB
Maple vncorbits7.maple 3.7KB

n=6

File format Download File size
graph6 vncorbits6.g6 55B
g6 with data vncorbits6.data 309B
ANF vncorbits6.anf 206B
Magma vncorbits6.magma 543B
Maple vncorbits6.maple 1.2KB

n=5

File format Download File size
graph6 vncorbits5.g6 16B
g6 with data vncorbits5.data 96B
ANF vncorbits5.anf 58B
Magma vncorbits5.magma 173B
Maple vncorbits5.maple 364B

n=4

File format Download File size
graph6 vncorbits4.g6 6B
g6 with data vncorbits4.data 38B
ANF vncorbits4.anf 20B
Magma vncorbits4.magma 74B
Maple vncorbits4.maple 140B

n=3

File format Download File size
graph6 vncorbits3.g6 3B
g6 with data vncorbits3.data 17B
ANF vncorbits3.anf 7B
Magma vncorbits3.magma 33B
Maple vncorbits3.maple 53B

n=2

File format Download File size
graph6 vncorbits2.g6 3B
g6 with data vncorbits2.data 15B
ANF vncorbits2.anf 4B
Magma vncorbits2.magma 27B
Maple vncorbits2.maple 40B

n=1

File format Download File size
graph6 vncorbits1.g6 2B
g6 with data vncorbits1.data 12B
ANF vncorbits1.anf 2B
Magma vncorbits1.magma 22B
Maple vncorbits1.maple 31B

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References

[1] L.E. Danielsen. On Self-Dual Quantum Codes, Graphs, and Boolean Functions, Master's thesis, Department of Informatics, University of Bergen, Norway, March 2005.

[2] A.R. Calderbank, E.M. Rains, P.M. Shor and N.J.A. Sloane. Quantum error correction via codes over GF(4). IEEE Transactions on Information Theory 44(4), pp. 1369-1387, 1998.

[3] A. Bouchet. Graphic presentations of isotropic systems. J. Combin. Theory Ser. B 45(1) 58-76, 1988.

[4] D. Schlingemann. Stabilizer codes can be realized as graph codes. Quantum Inf. Comput. 2(4) 307-323, 2002.

[5] D.G. Glynn. On self-dual quantum codes and graphs. Submitted to Electronic Journal of Combinatorics. Preprint, 2002.

[6] D.G. Glynn, T.A. Gulliver, J.G. Maks and M.K. Gupta. The Geometry of Additive Quantum Codes. Submitted to Springer-Verlag, 2004.

[7] M. Hein, J. Eisert, H.J. Briegel. Multi-party entanglement in graph states. Phys. Rev. A 69(6), 2004.

[8] M. Van den Nest, J. Dehaene and B. De Moor. Graphical description of the action of local Clifford transformations on graph states. Phys. Rev. A 69(2), 2004.

[9] G. Höhn. Self-dual codes over the Kleinian four group. Mathematische Annalen 327, pp. 227-255, 2003.

[10] P. Gaborit P, W.C. Huffman, J.-L. Kim, and V. Pless V. On additive GF(4) codes. In "Codes and Association Schemes", vol. 56 of DIMACS Ser. Discrete Math. Theoret. Comput. Sci., pp. 135-149. Amer. Math. Soc., 2001.

[11] C. Bachoc and P. Gaborit. On extremal additive F4 codes of length 10 to 18. J. Théor. Nombres Bordeaux 12(2) 255-271, 2000.

[12] L.E. Danielsen and M.G. Parker. Spectral orbits and peak-to-average power ratio of Boolean functions with respect to the {I,H,N}^n transform, January 2005. To appear in the proceedings of Sequences and Their Applications, SETA'04, Lecture Notes in Computer Science, Springer-Verlag.

[13] L.E. Danielsen, T.A. Gulliver and M.G. Parker. Aperiodic propagation criteria for Boolean functions. Submitted to Inform. Comput., October 2004.

[14] Lars Eirik Danielsen and Matthew G. Parker: On the classification of all self-dual additive codes over GF(4) of length up to 12, April 2005. Submitted to Journal of Combinatorial Theory, Series A.