The generator matrix
1 1 1 1 1 1 1 1 1 1
0 X 0 0 0 X 0 0 X X
0 0 X 0 0 X 0 X X 0
0 0 0 X 0 X X X 0 0
0 0 0 0 X X X 0 0 X
generates a code of length 10 over Z2[X]/(X^2) who´s minimum homogenous weight is 8.
Homogenous weight enumerator: w(x)=1x^0+15x^8+32x^10+15x^12+1x^20
The gray image is a linear code over GF(2) with n=20, k=6 and d=8.
As d=8 is an upper bound for linear (20,6,2)-codes, this code is optimal over Z2[X]/(X^2) for dimension 6.
This code was found by Heurico 1.16 in 0.000325 seconds.