The generator matrix
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X X X X X X X 0 1 1 1 0 0 0 0 0 0 1
0 X 0 0 0 X X X 0 0 0 X 0 X X X 0 0 X 0 X X X X 0 0 0 X X X 0 0 0 X
0 0 X 0 X X X 0 0 0 X X X X 0 0 0 X X X X 0 0 0 0 0 X 0 X X X X 0 X
0 0 0 X X 0 X X 0 X X 0 0 X X 0 X X 0 0 X X 0 0 0 X X X X 0 0 X X 0
generates a code of length 34 over Z2[X]/(X^2) who´s minimum homogenous weight is 34.
Homogenous weight enumerator: w(x)=1x^0+20x^34+6x^36+4x^38+1x^40
The gray image is a linear code over GF(2) with n=68, k=5 and d=34.
As d=34 is an upper bound for linear (68,5,2)-codes, this code is optimal over Z2[X]/(X^2) for dimension 5.
This code was found by Heurico 1.16 in 0.00778 seconds.