### 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 .

A col (or column) is a placement of three '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 '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 with finite slope occurs once in the part of some tet. Define the label of the first (top) row to be 0 , of the second row to , and of the bottom row to . 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 .

Associated to the col+tet pattern

 1 1 1
+
 1 1 1 1
=
 1 1 2 1 1 1
is the tetracode .

david joyner 2008-04-20