Doubly-Even Codes

The number of permutation equivalence classes of doubly-even binary linear codes is shown in the table below (counting only codes without zero columns). Mouse over the numbers in the table to see the codes (when feasible).

You can download individual N, k code lists here, but you may need to use an older version of Sage so that the sobj files load.

sage: L = load('16_08_de_codes.sobj')
sage: L[0]
[1 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0]
[0 1 0 0 0 0 0 0 1 1 1 1 1 1 0 1]
[0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 1]
[0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 1]
[0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 1]
[0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 1]
[0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 1]
[0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1]

Talk

The following machines have been used for carrying out these computations:

• Ohio Supercomputing Center's Glenn
• Greg Landweber's Mac Pro
• Tom Boothby's eight
• William Stein's sage.math

Let F denote the field containing two elements. A linear binary code of type [N, k] is a vector subspace C of F^N with dimension k. Using the standard basis, the weight wt(v) of a vector v in an F-vector space is simply the number of coordinates equal to one. Define a doubly-even code to be a linear binary code with the constraint that every vector has weight divisible by 4. We use Gaborit's formulas for the number of distinct doubly-even codes.

An interesting fact is that doubly-even codes are guaranteed to be self-orthogonal, under the standard inner product < , >. We can express the dot product of two vectors, modulo 2, as the number of coordinates where both vectors are one. We can express this in terms of weight:

For example,

x =	1	1	1	1	0	0
y =	0	0	1	1	1	1
x + y =	1	1	0	0	1	1.