# 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,2 ^{n},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}*-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

^{n}*{I,H,N}*and other symmetries. For quadratic and nonquadratic Boolean functions we are interested in PAR

^{n}_{IHN}, the

*Peak-to-Average power Ratio*with respect to the

*{I,H,N}*transform, and APC, the

^{n}*Aperiodic Propagation Criteria*[13]. Both PAR

_{IHN}and APC of a Boolean function are related to the distance of the corresponding quantum code.

(See also David Glynn's web page.)

^ TOP# Tables

Number of inequivalent indecomposable (i_{n}) and possibly decomposable
(t_{n}) self-dual quantum codes:

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

i_{n} | 1 | 1 | 1 | 2 | 4 | 11 | 26 | 101 | 440 | 3,132 | 40,457 | 1,274,068 |

t_{n} | 1 | 2 | 3 | 6 | 11 | 26 | 59 | 182 | 675 | 3,990 | 45,144 | 1,323,363 |

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\n | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
---|---|---|---|---|---|---|---|---|---|---|---|

2 | 1 | 1 | 2 | 3 | 9 | 22 | 85 | 363 | 2,436 | 26,750 | 611,036 |

3 | 1 | 1 | 4 | 11 | 69 | 576 | 11,200 | 467,513 | |||

4 | 1 | 5 | 8 | 120 | 2,506 | 195,455 | |||||

5 | 1 | 63 | |||||||||

6 | 1 |

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

^ TOP# 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:

- graph in nauty's graph6 format
- size of orbit
- distance of code
- weight distribution of code

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

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:

- length of code
- minimum distance of code
- decomposable or indecomposable code? (Value: D or I)
- type I or type II code? (Value: 1 or 2)
- weight distribution of code
- size of code automorphism group
- LC orbit size
- size of maximum independent set in LC orbit
- graph in nauty's graph6 format

Download | File size |
---|---|

thebigselfdualfile.txt.gz | 14MB |

# 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 |

# 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.

- Also available from:

arXiv:quant-ph/0503236

http://www.ub.uib.no/elpub/2005/h/413001/

[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 F_{4} 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.

- Also available from: arXiv:math.CO/0504522