Last update 2017/03/21

Update 2011/6/29

Update 2011/6/29

A triply even code is a binary code in which the weight of every codeword is divisible by 8.

In this page, a triply even code is a triply even code which contains the all-ones vector.

We thank Dr. Michael Kiermaier whose comment prompted us to clarify this additional assumption we have made in the classification.

In this page, a triply even code is a triply even code which contains the all-ones vector.

We thank Dr. Michael Kiermaier whose comment prompted us to clarify this additional assumption we have made in the classification.

Dimension | Number of codes |
---|---|

1 | 1 |

2 | 3 |

3 | 10 |

4 | 35 |

5 | 136 |

6 | 458 |

7 | 1162 |

8 | 1910 |

9 | 1960 |

10 | 1247 |

11 | 520 |

12 | 159 |

13 | 39 |

14 | 6 |

15 | 1 |

Dimension | ID | remark |
---|---|---|

9 | 1 | The triangular code |

8 | 1 | |

7 | 3 |

The paper of this project "On triply even binary codes" is available at

arxiv [1012.4134].

The magma script to obtain this result is

classification.tex

This file is excutable on Magma as follows:

> magma classification.tex

It is available on Latex as well with the file:

triply-even.tex.

One can obtain the script with a reader friendly format and comments by executing command

> latex triply-even.tex

The following files give a complete list of triply even codes of length 48 up to equivalence.

The generator matrices are given by the heximal form in

File of Generator Matrix [Heximal form |
Matrix form |
Magma form].

Each code is described by the form

<Dimension, Code Id, [ Generators ]>

For example, the 132^{nd} code of dimension 5 is descrebed by the heximal form and the matrix form
respectively as follows:

<5, 132, [ 0x9669, 0xFFFFFFFFAAAA, 0xFFFFFFFFCCCC, 0xFFFFFFFFF0F0, 0xFF00 ]>,

<5, 132, [48, 5, 8] [100101100110100100000000000000000000000000000000] [010101010101010111111111111111111111111111111111] [001100110011001111111111111111111111111111111111] [000011110000111111111111111111111111111111111111] [000000001111111100000000000000000000000000000000]>,

The following files give the composition factors of automorphism groups of codes.

Dimension = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15

The following files give the weight enumerators of codes.

Each weight enumerator is described by a list of pair of the degree
and coefficient of each monomial. For example, <3, 4,
[ <0, 1>, <24, 6>, <48, 1> ]>
indicates \(1+6x^{24}+x^{48}\).

Dimension = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15

The list of codes constructed by doubling is given by File of the list of doublings.

The format is as follows

\(\langle k, [
\langle\mathrm{id}_1,\mathrm{id}'_1\rangle,
\langle\mathrm{id}_2,\mathrm{id}'_2\rangle,\ldots,
\langle\mathrm{id}_n,\mathrm{id}'_n\rangle]\rangle\)

\(\mathrm{id}_i\) indecates the code \(C_{\mathrm{id}_i}\)
of dimension \(k\) which is equivalent to
a code of the form \(\{(c \mid c) \mid c \in C\}\)

and \(\mathrm{id}'_i\) indecates the code \(C'_{\mathrm{id}'_i}\)
of dimension \(k+1\) which is equivalent to \(C_{\mathrm{id}_i} +
\langle (0,\ldots,0,1,\ldots,1)\rangle\).

The list of codes constructed by extended doubling is given by File of the list of extended doublings.

The format is as follows

\(\langle k,\mathrm{id}\rangle, [
\langle k_1,\mathrm{id}_1\rangle,
\langle k_2,\mathrm{id}_2\rangle,\ldots,
\langle k_m,\mathrm{id}_m\rangle]\rangle\)

where
\(\langle k_i,\mathrm{id}_i\rangle\)
indecates the code \(C_{\mathrm{id}_i}\)
of dimension \(k_i\) which is equivalent to
a code of the form \(\{(c \mid c) \mid c \in C\}\)

and \(C_{\mathrm{id}}\)
is an exteded doubling code constructed from each code \(C_{\mathrm{id}_i}\),

which is equivalent to \(C_{\mathrm{id}_i} + \langle (0 \mid D )\rangle\)
where \(C_{\mathrm{id}_i} =\{(c\mid c) \mid c \in C\}\) and \(D=\mathrm{Rad}\; C\).

File of the list of non-embeddable codes gives the list of codes non-embeddable in doublings.

The list of decomposable codes is given in File of decomposable codes.

Each component is given in File of components of decomposable codes.

An entry \(\langle k,\mathrm{id}_1, \mathrm{id}_2\rangle\) indecats that the code \(C_{\mathrm{id}_2}\) of dimension \(k+1\)

can be obtained from the code \(C_{\mathrm{id}_1}\) of dimension \(k\) by attaching some weight 8 vector.

The following codes are the maximal triply even codes.
<15,1> : Three copies of \(\mathrm{RM}(1,4)\)
<14,1> : Direct sum of \(\mathrm{RM}(1,4)\) and doubling of \(d_{16}^{+}\).
<13,1> : Doubling of \(g_{24}\) of order of Aut is 1002795171840.
<13,2> : Doubling of \((d_{10}e_7^2)^{+}\) of order of Aut is 443925135360.
<13,3> : Doubling of \(d_{24}^{+}\) of order of Aut is 12054469961318400.
<13,4> : Doubling of \(d_{12}^{2+}\) of order of Aut is 4348654387200.
<13,5> : Doubling of \(d_4^{6+}\) of order of Aut is 36238786560.
<13,6> : Doubling of \(d_6^{4+}\) of order of Aut is 32614907904.
<13,7> : Doubling of \(d_8^{3+}\) of order of Aut is 173946175488.
<9,1> : The code induced from the triangular graph.