The ``col/tet'' construction

In fact, this only constructs the codewords of weight 6, but since they generate the code, we can use them to compute a generating matrix for $C_{12}$ .

A col (or column) is a placement of three $1$ 's in a column of the array (a blank space represents a 0 ):

1
1
1

	1
	1
	1

		1
		1
		1

			1
			1
			1

A tet (or tetrad) is a placement of 4 $1$ 's having entries corresponding (as explained below) to a tetracode.

1	1	1	1

1
	1	1	1

1
	1	1	1

0

0

0

0

0

1

1

1

0

2

2

2

	1
1		1
			1

			1
1	1
		1

		1
1			1
	1

1

0

1

2

1

1

2

0

1

2

0

1

	1
			1
1		1

		1
	1
1			1

			1
		1
1	1

2

0

2

1

2

1

0

2

2

2

1

0

Each line in $\mathbb{F}_3^2$ with finite slope occurs once in the $3\times 3$ part of some tet. Define the label of the first (top) row to be 0 , of the second row to $-1$ , and of the bottom row to $1$ . The odd man out for a column is the label of the row corresponding to the non-zero digit in that column; if the column has no non-zero digit then the odd man out is a ``?''. Thus the tetracode words associated in this way to these patterns are the odd men out for the tets.

Example 6.6.7   Associated to the col-col pattern

1
1
1

-

	1
	1
	1

=

1	2
1	2
1	2

is the tetracode $(0, 0, ?, ?)$ .

Associated to the col+tet pattern

	1
	1
	1

+

1
	1	1	1

=

1	1
	2	1	1
	1

is the tetracode $(0, 1, 1, 1)$ .