The generator matrix
1 1 1 1 X 1
0 X 0 0 X 0
0 0 X 0 X 0
0 0 0 X X 0
generates a code of length 6 over Z2[X]/(X^2) who´s minimum homogenous weight is 4.
Homogenous weight enumerator: w(x)=1x^0+6x^4+8x^5+4x^6+8x^7+1x^8+4x^10
The gray image is a linear code over GF(2) with n=12, k=5 and d=4.
As d=4 is an upper bound for linear (12,5,2)-codes, this code is optimal over Z2[X]/(X^2) for dimension 5.
This code was found by Heurico 1.16 in 7.51e-005 seconds.