# Introduction

This is a database of self-dual quantum codes over GF(3), GF(4), and GF(5). These can also be interpreted as classical additive codes over GF(9), GF(16), and GF(25) that are self-dual with respect to the Hermitian trace inner product. For a classification of binary quantum codes (self-dual additive codes over GF(4)), see the Database of Self-Dual Quantum Codes.

These codes were classified by using the fact that they can be represented as weighted graphs, and that orbits of graphs under generalized local complementation correspond to equivalence classes of codes. For details, see the paper [1].

The classification of self-dual additive codes over GF(9) has later been extended from length 8 to length 10, using a different approach. All optimal codes over GF(9) of length 11 and 12 have also been classified. For details, see [2].

^ TOP# Tables

Total number of self-dual quantum codes over GF(m) (self-dual additive codes over GF(m^{2})) of length *n*:

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

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

3 | 1 | 2 | 3 | 7 | 13 | 39 | 121 | 817 | 18,525 | 2,822,779 | ||

4 | 1 | 2 | 3 | 7 | 14 | 44 | ||||||

5 | 1 | 2 | 3 | 7 | 15 | 58 |

Number of indecomposable self-dual quantum codes over GF(3) (self-dual additive codes over GF(9)) of length *n* and distance *d*:

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

2 | 1 | 1 | 2 | 4 | 15 | 51 | 388 | 6,240 | 418,088 | ? | ? |

3 | 1 | 1 | 5 | 20 | 194 | 6,975 | 893,422 | ? | ? | ||

4 | 1 | 2 | 77 | 4,370 | 1,487,316 | ? | ? | ||||

5 | 4 | 4,577 | 56,005,876 | ? | |||||||

6 | 1 | 6,493 | |||||||||

All | 1 | 1 | 3 | 5 | 21 | 73 | 659 | 17,589 | 2,803,404 | ? | ? |

Number of indecomposable self-dual quantum codes over GF(4) (self-dual additive codes over GF(16)) of length *n* and distance *d*:

d\n | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|

2 | 1 | 1 | 2 | 4 | 16 |

3 | 1 | 2 | 6 | ||

4 | 3 | ||||

5 | |||||

6 | |||||

All | 1 | 1 | 3 | 6 | 25 |

Number of indecomposable self-dual quantum codes over GF(5) (self-dual additive codes over GF(25)) of length *n* and distance *d*:

d\n | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|

2 | 1 | 1 | 2 | 4 | 21 |

3 | 1 | 3 | 11 | ||

4 | 6 | ||||

5 | |||||

6 | |||||

All | 1 | 1 | 3 | 7 | 38 |

See the papers [1,2] for more tables and information.

^ TOP# Files

There is one file for each alphabet. The file for m=3 contains both indecomposable and decomposable codes, and the records have the following format:

- Length of code
- Decomposable or Indecomposable code? (Value: D or I)
- Minimum distance of code
- Weight distribution of code
- Size of automorphism group of code
- Weighted graph representation of code (Example: the graph "0-3,2*1-3,3*2-3" has edge {0,3} with weight 1, edge {1,3} with weight 2, and edge {2,3} with weight 2.) For how to map this graph to a code, see [2].

The files for m=4 and m=5 contain indecomposable codes only, and the records have the following format:

- Length of code
- Minimum distance of code
- LC orbit size
- Weighted graph representation of code (Example: the graph "03,2*13,3*23" has edge {0,3} with weight 1, edge {1,3} with weight 2, and edge {2,3} with weight 2.)
- In the case m=4, the weighs on the graph are over GF(4), and should be interpreted as 0=0, 1=1, 2=a, 3=a+1, where a is a primitive element of GF(4).

m | Download | |
---|---|---|

3 | selfdualcodes3.txt.bz2 (32 MB) | Contains all codes of length up to 10, and all optimal codes of length 12. (The optimal codes of length 11 are to numerous to include here, contact me if you need them.) |

4 | selfdualcodes4.txt | |

5 | selfdualcodes5.txt |

# References

[1] Lars Eirik Danielsen.
Graph-based classification of self-dual additive codes over finite fields.
*Adv. Math. Commun.* 3(4), 329–348, 2009.
(doi)

[2] Lars Eirik Danielsen. Classification of Hermitian self-dual additive codes over GF(9). Jun. 2011. (Submitted for publication) (arXiv)