Magma URL:

http://magma.maths.usyd.edu.au/calc/

Magma code sample:

> V24 := VectorSpace(FiniteField(2), 24);
> G := sub< V24 |  
> [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1],
> [1,0,0,0,0,0,0,1,0,0,0,1,0,1,1,1,0,0,1,0,0,1,0,0]>;       
> Dimension(G);

Draft of code for exercise of 9/11, 2022,
in format for checking patterns —

Details of patterns within the array of 24 patterns,
top to bottom, left to right —

Column 1:

> [0,0,0,0,0,1,
    0,1,0,0,1,1,
    1,0,0,1,0,1,
    1,1,0,1,1,1],
> [0,1,0,0,0,0,
    0,1,1,0,0,1,
    1,1,0,1,0,0,
    1,1,1,1,0,1],
> [0,1,1,0,1,0,
    0,0,1,0,0,0,
    1,1,1,1,1,0,
    1,0,1,1,0,0],   
> [0,0,1,0,1,1,
    0,0,0,0,1,0,
    1,0,1,1,1,1,
    1,0,0,1,1,0],

Column 2:

> [1,0,0,0,0,0,
    1,1,0,0,1,0,
    1,0,1,0,0,1,
    1,1,1,0,1,1],
> [1,1,0,1,0,0,
    0,1,0,0,0,0,
    1,1,1,1,0,1,
    0,1,1,0,0,1],
> [0,1,0,1,1,0,
    0,0,0,1,0,0,
    0,1,1,1,1,1,
    0,0,1,1,0,1],
> [0,0,0,0,1,0,
    1,0,0,1,1,0,
    0,0,1,0,1,1,
    1,0,1,1,1,1],

Column 3:

> [1,0,1,1,0,0,
    1,1,1,1,1,0,
    0,0,1,0,0,0,
    0,1,1,0,1,0],
> [1,1,1,1,0,1,
    1,1,0,1,0,0,
    0,1,1,0,0,1,
    0,1,0,0,0,0]
> [1,1,0,1,1,1,
    1,0,0,1,0,1,
    0,1,0,0,1,1,
    0,0,0,0,0,1],
> [1,0,0,1,1,0,
    1,0,1,1,1,1,
    0,0,0,0,1,0,
    0,0,1,0,1,1],

Nondraft of above code for columns 1-3
and commands:

> V24 := VectorSpace(FiniteField(2), 24);
> G := sub< V24 |  
> [0,0,0,0,0,1,0,1,0,0,1,1,1,0,0,1,0,1,1,1,0,1,1,1],
> [0,1,0,0,0,0,0,1,1,0,0,1,1,1,0,1,0,0,1,1,1,1,0,1],
> [0,1,1,0,1,0,0,0,1,0,0,0,1,1,1,1,1,0,1,0,1,1,0,0],
> [0,0,1,0,1,1,0,0,0,0,1,0,1,0,1,1,1,1,1,0,0,1,1,0],
> [1,0,0,0,0,0,1,1,0,0,1,0,1,0,1,0,0,1,1,1,1,0,1,1],
> [1,1,0,1,0,0,0,1,0,0,0,0,1,1,1,1,0,1,0,1,1,0,0,1],
> [0,1,0,1,1,0,0,0,0,1,0,0,0,1,1,1,1,1,0,0,1,1,0,1],
> [0,0,0,0,1,0,1,0,0,1,1,0,0,0,1,0,1,1,1,0,1,1,1,1],
> [1,0,1,1,0,0,1,1,1,1,1,0,0,0,1,0,0,0,0,1,1,0,1,0],
> [1,1,1,1,0,1,1,1,0,1,0,0,0,1,1,0,0,1,0,1,0,0,0,0],
> [1,1,0,1,1,1,1,0,0,1,0,1,0,1,0,0,1,1,0,0,0,0,0,1],
> [1,0,0,1,1,0,1,0,1,1,1,1,0,0,0,0,1,0,0,0,1,0,1,1]>;
> Dimension(G);


Column 4:

> [0,0,1,1,0,1,
    0,1,1,1,1,1,
    0,0,0,1,0,0,
    0,1,0,1,1,0],
> [0,1,1,0,0,1,
     1,1,1,1,0,1,
     0,1,0,0,0,0,
     1,1,0,1,0,0],
> [1,1,1,0,1,1,
    1,0,1,0,0,1,
    1,1,0,0,1,0,
    1,0,0,0,0,0],
> [1,0,1,1,1,1,
    0,0,1,0,1,1,
    1,0,0,1,1,0,
    0,0,0,0,1,0],

Column 5:

> [1,0,0,1,0,1,
    0,0,0,0,0,1,
    1,1,0,1,1,1,
    0,1,0,0,1,1],
> [0,0,0,1,0,0,
    0,0,1,1,0,1,
    0,1,0,1,1,0,
    0,1,1,1,1,1],
> [0,0,1,0,0,0,
    1,0,1,1,0,0,
    0,1,1,0,1,0,
    1,1,1,1,1,0],
> [1,0,1,0,0,1,
    1,0,0,0,0,0,
    1,1,1,0,1,1,
    1,1,0,0,1,0],

Column 6:

> [0,1,0,0,1,1,
    1,1,0,1,1,1,
    0,0,0,0,0,1,
    1,0,0,1,0,1],
> [1,1,0,0,1,0,
    1,1,1,0,1,1,
    1,0,0,0,0,0,
    1,0,1,0,0,1],
> [1,1,1,1,1,0,
    0,1,1,0,1,0,
    1,0,1,1,0,0,
    0,0,1,0,0,0],
> [0,1,1,1,1,1,
    0,1,0,1,1,0,
    0,0,1,1,0,1,
    0,0,0,1,0,0],

Columns 4, 5, 6 in Magma-entry form:

Column 4:

> [0,0,1,1,0,1,0,1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,1,0],
> [0,1,1,0,0,1,1,1,1,1,0,1,0,1,0,0,0,0,1,1,0,1,0,0],
> [1,1,1,0,1,1,1,0,1,0,0,1,1,1,0,0,1,0,1,0,0,0,0,0],
> [1,0,1,1,1,1,0,0,1,0,1,1,1,0,0,1,1,0,0,0,0,0,1,0],

Column 5:

> [1,0,0,1,0,1,0,0,0,0,0,1,1,1,0,1,1,1,0,1,0,0,1,1],
> [0,0,0,1,0,0,0,0,1,1,0,1,0,1,0,1,1,0,0,1,1,1,1,1],
> [0,0,1,0,0,0,1,0,1,1,0,0,0,1,1,0,1,0,1,1,1,1,1,0],
> [1,0,1,0,0,1,1,0,0,0,0,0,1,1,1,0,1,1,1,1,0,0,1,0],

Column 6:

> [0,1,0,0,1,1,1,1,0,1,1,1,0,0,0,0,0,1,1,0,0,1,0,1],
> [1,1,0,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0,1,0,1,0,0,1],
> [1,1,1,1,1,0,0,1,1,0,1,0,1,0,1,1,0,0,0,0,1,0,0,0],
> [0,1,1,1,1,1,0,1,0,1,1,0,0,0,1,1,0,1,0,0,0,1,0,0],

Dimension of space generated by columns 4, 5, 6 —

> V24 := VectorSpace(FiniteField(2), 24);
> G := sub< V24 |
> [0,0,1,1,0,1,0,1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,1,0],
> [0,1,1,0,0,1,1,1,1,1,0,1,0,1,0,0,0,0,1,1,0,1,0,0],
> [1,1,1,0,1,1,1,0,1,0,0,1,1,1,0,0,1,0,1,0,0,0,0,0],
> [1,0,1,1,1,1,0,0,1,0,1,1,1,0,0,1,1,0,0,0,0,0,1,0],
> [1,0,0,1,0,1,0,0,0,0,0,1,1,1,0,1,1,1,0,1,0,0,1,1],
> [0,0,0,1,0,0,0,0,1,1,0,1,0,1,0,1,1,0,0,1,1,1,1,1],
> [0,0,1,0,0,0,1,0,1,1,0,0,0,1,1,0,1,0,1,1,1,1,1,0],
> [1,0,1,0,0,1,1,0,0,0,0,0,1,1,1,0,1,1,1,1,0,0,1,0],
> [0,1,0,0,1,1,1,1,0,1,1,1,0,0,0,0,0,1,1,0,0,1,0,1],
> [1,1,0,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0,1,0,1,0,0,1],
> [1,1,1,1,1,0,0,1,1,0,1,0,1,0,1,1,0,0,0,0,1,0,0,0],
> [0,1,1,1,1,1,0,1,0,1,1,0,0,0,1,1,0,1,0,0,0,1,0,0]>;
> Dimension(G);

Magma calculation of dimension of space generated
by all 24 motifs from 1981 —

> V24 := VectorSpace(FiniteField(2), 24);
> G := sub< V24 |
> [0,0,0,0,0,1,0,1,0,0,1,1,1,0,0,1,0,1,1,1,0,1,1,1],
> [0,1,0,0,0,0,0,1,1,0,0,1,1,1,0,1,0,0,1,1,1,1,0,1],
> [0,1,1,0,1,0,0,0,1,0,0,0,1,1,1,1,1,0,1,0,1,1,0,0],
> [0,0,1,0,1,1,0,0,0,0,1,0,1,0,1,1,1,1,1,0,0,1,1,0],
> [1,0,0,0,0,0,1,1,0,0,1,0,1,0,1,0,0,1,1,1,1,0,1,1],
> [1,1,0,1,0,0,0,1,0,0,0,0,1,1,1,1,0,1,0,1,1,0,0,1],
> [0,1,0,1,1,0,0,0,0,1,0,0,0,1,1,1,1,1,0,0,1,1,0,1],
> [0,0,0,0,1,0,1,0,0,1,1,0,0,0,1,0,1,1,1,0,1,1,1,1],
> [1,0,1,1,0,0,1,1,1,1,1,0,0,0,1,0,0,0,0,1,1,0,1,0],
> [1,1,1,1,0,1,1,1,0,1,0,0,0,1,1,0,0,1,0,1,0,0,0,0],
> [1,1,0,1,1,1,1,0,0,1,0,1,0,1,0,0,1,1,0,0,0,0,0,1],
> [1,0,0,1,1,0,1,0,1,1,1,1,0,0,0,0,1,0,0,0,1,0,1,1],
> [0,0,1,1,0,1,0,1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,1,0],
> [0,1,1,0,0,1,1,1,1,1,0,1,0,1,0,0,0,0,1,1,0,1,0,0],
> [1,1,1,0,1,1,1,0,1,0,0,1,1,1,0,0,1,0,1,0,0,0,0,0],
> [1,0,1,1,1,1,0,0,1,0,1,1,1,0,0,1,1,0,0,0,0,0,1,0],
> [1,0,0,1,0,1,0,0,0,0,0,1,1,1,0,1,1,1,0,1,0,0,1,1],
> [0,0,0,1,0,0,0,0,1,1,0,1,0,1,0,1,1,0,0,1,1,1,1,1],
> [0,0,1,0,0,0,1,0,1,1,0,0,0,1,1,0,1,0,1,1,1,1,1,0],
> [1,0,1,0,0,1,1,0,0,0,0,0,1,1,1,0,1,1,1,1,0,0,1,0],
> [0,1,0,0,1,1,1,1,0,1,1,1,0,0,0,0,0,1,1,0,0,1,0,1],
> [1,1,0,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0,1,0,1,0,0,1],
> [1,1,1,1,1,0,0,1,1,0,1,0,1,0,1,1,0,0,0,0,1,0,0,0],
> [0,1,1,1,1,1,0,1,0,1,1,0,0,0,1,1,0,1,0,0,0,1,0,0]>;
> Dimension(G);

The result is 8 dimensions, instead of the Golay code's 12.  

Trial extensions of the set of 24 basis vectors . . .

> 0,0,1,1,0,0,
  0,0,1,1,0,0,
  0,0,1,1,0,0,
  0,0,1,1,0,0],
as
> [0,0,1,1,0,0,0,0,1,1,0,0,0,0,1,1,0,0,0,0,1,1,0,0],

> [1,0,0,0,0,1,
  1,0,0,0,0,1,
  1,0,0,0,0,1,
  1,0,0,0,0,1],
as
> [1,0,0,0,0,1,1,0,0,0,0,1,1,0,0,0,0,1,1,0,0,0,0,1],

> [1,1,1,1,1,1,
  1,1,1,1,1,1,
  0,0,0,0,0,0,
  0,0,0,0,0,0],
as
> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0],

> [1,1,1,,0,0,0,
  1,1,1,,0,0,0,
  1,1,1,,0,0,0,
  1,1,1,,0,0,0,],
as
> [1,1,1,,0,0,0,1,1,1,,0,0,0,1,1,1,,0,0,0,1,1,1,,0,0,0,],

Set of 4 symmetric extension vectors . . .

> [0,0,1,1,0,0,0,0,1,1,0,0,0,0,1,1,0,0,0,0,1,1,0,0],
> [1,0,0,0,0,1,1,0,0,0,0,1,1,0,0,0,0,1,1,0,0,0,0,1],
> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0],
> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0],

Adding these four as basis vectors increases the
dimension of the resulting space from 8 to 11.

Another trial symmetric basis extension vector . . .
the "left brick" of R. T. Curtis . . .

> [1,1,0,0,0,0,
  1,1,0,0,0,0,
  1,1,0,0,0,0,
  1,1,0,0,0,0],
as
> [1,1,0,0,0,0,1,1,0,0,0,0,1,1,0,0,0,0,1,1,0,0,0,0],

Revised set of four added basis vectors . . .
Left brick, middle brick, top half, left half . . .

> [1,1,0,0,0,0,1,1,0,0,0,0,1,1,0,0,0,0,1,1,0,0,0,0],
> [0,0,1,1,0,0,0,0,1,1,0,0,0,0,1,1,0,0,0,0,1,1,0,0],
> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0],
> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]>;

These four bring the resulting space from 8 to 12 dimensions.

— Steven H. Cullinane, Mon., Sept. 12, 2022, 3:55 PM ET