The generator matrix
1 1 1 1 1 1 1 1
0 X 0 0 0 X X X
0 0 X 0 X X X 0
0 0 0 X X 0 X X
generates a code of length 8 over Z2[X]/(X^2) who´s minimum homogenous weight is 8.
Homogenous weight enumerator: w(x)=1x^0+30x^8+1x^16
The gray image is a linear code over GF(2) with n=16, k=5 and d=8.
As d=8 is an upper bound for linear (16,5,2)-codes, this code is optimal over Z2[X]/(X^2) for dimension 5.
This code was found by Heurico 1.16 in 0.000157 seconds.