# Introduction

This is a database of objects that have several interpretations:

- Equivalence classes of self-dual additive codes over GF(4). [1,3,13,14]
- Equivalence classes of self-dual codes over the Kleinian four group. [12]
- Orbits of simple undirected graphs under local complementation. [1,4,7,8,10,11]
- Equivalence classes of isotropic systems. [4]
- Equivalence classes of zero-dimensional quantum codes. [5–11]
- LU-equivalence classes of quantum graph states. [9,10,11]
- Equivalence classes of quadratic Boolean functions under certain transformations. [15]

For a more detailed description of the results, see the paper [1], or my Master's thesis [2].

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 the Hermitian trace inner product [3]. 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* [4–11].
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) [4,7,8,10,11].

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. [3]
(up to *n*=5), by Hein et al. [9] (up to *n*=7),
by Höhn [12] (up to *n*=7) and by Glynn et al. [8] (up to *n*=9).
For higher lengths, only optimal codes have been classified [13,14].
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 *decomposable* code can be expressed as the direct sum of two smaller codes. Indecomposable codes
correspond to connected graphs. The decomposable codes can easily be constructed by combining
indecomposable codes of shorter length. A self-dual additive code over GF(4) is of *Type II*
if all codewords have even weight. We have also classified all Type II (14,2^{14},6) codes.

# Tables

Number of indecomposable (i_{n}) and total number
(t_{n}) of 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.

Total number of codes of length *n* and distance *d*:

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

1 | 1 | 1 | 2 | 3 | 6 | 11 | 26 | 59 | 182 | 675 | 3,990 | 45,144 |

2 | 1 | 1 | 3 | 4 | 13 | 29 | 107 | 416 | 2,618 | 27,445 | 615,180 | |

3 | 1 | 1 | 4 | 11 | 69 | 577 | 11,202 | 467,519 | ||||

4 | 1 | 5 | 8 | 120 | 2,506 | 195,456 | ||||||

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

6 | 1 | |||||||||||

All | 1 | 2 | 3 | 6 | 11 | 26 | 59 | 182 | 675 | 3,990 | 45,144 | 1,323,363 |

Total number of Type II codes of length *n* and distance *d*:

d\n | 2 | 4 | 6 | 8 | 10 | 12 | 14 |
---|---|---|---|---|---|---|---|

2 | 1 | 2 | 5 | 18 | 109 | 2,285 | ? |

4 | 1 | 3 | 19 | 793 | ? | ||

6 | 1 | 1,020 | |||||

All | 1 | 2 | 6 | 21 | 128 | 3,079 | ≥1,727,942 |

See the paper [1] for more tables and information.

^ TOP# File formats

We make our data available in two formats.
Generator matrices for self-dual additive codes over GF(4) are stored in files that can be read by Magma.
Graphs are stored in nauty's graph6 format. This
compact representation can be transformed into Magma and Maple format by using the nauty
package. The graph6 files can be used as input to all the utilities in the
nauty package.
Note that the graph with adjacency matrix *Γ* represents the code with generator matrix *Γ + ωI*.

# Files

## One big file

You can download one big file containing all data. The records have the following format:

- 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 |
---|

thebigselfdualfile.txt.gz |

## Generator matrices in Magma format

These files contain all codes of length *n*. For length 14, only extremal Type II codes are included.

n | Download |
---|---|

1 | codes1.mag |

2 | codes2.mag |

3 | codes3.mag |

4 | codes4.mag |

5 | codes5.mag |

6 | codes6.mag |

7 | codes7.mag |

8 | codes8.mag |

9 | codes9.mag |

10 | codes10.mag |

11 | codes11.mag |

12 | codes12.mag.gz |

14 | codes14.mag |

## Graphs in nauty's format

These files contain only connected graphs (corresponding to indecomposable codes). For length 14, only graphs corresponding to indecomposable extremal Type II codes are included.

n | Download |
---|---|

1 | vncorbits1.g6 |

2 | vncorbits2.g6 |

3 | vncorbits3.g6 |

4 | vncorbits4.g6 |

5 | vncorbits5.g6 |

6 | vncorbits6.g6 |

7 | vncorbits7.g6 |

8 | vncorbits8.g6 |

9 | vncorbits9.g6 |

10 | vncorbits10.g6 |

11 | vncorbits11.g6 |

12 | vncorbits12.g6.gz |

14 | vncorbits14.g6 |

# References

[1] Lars Eirik Danielsen and Matthew G. Parker:
On the classification of all self-dual additive codes over GF(4) of length up to 12,
*J. Combin. Theory Ser. A* **113**(7), pp. 1351–1367, 2006.
(Also available from: arXiv:math.CO/0504522.)

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

[3] 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.

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

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

[6] M. Grassl, A. Klappenecker, M. Rötteler. Graphs, quadratic forms, and quantum codes.
In *Proc. IEEE Int. Symp. Inform. Theory* (ISIT 2002), 2002, p. 45.

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

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

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

[10] 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.

[11] M. Van den Nest. Local equivalence of stabilizer states and codes. Ph.D. thesis, K. U. Leuven, Leuven, Belgium, May 2005.

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

[13] 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.

[14] 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.

[15] 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.
In *Sequences and Their Applications – SETA 2004*, *Lecture Notes in Computer Science*,
volume 3486, pp. 373-388, Springer-Verlag, Berlin, 2005.