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 132nd 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.