The generator matrix

 1  0  0  0  0  0  0  1  1  1  1
 0  1  0  0  0  0  0  1  X  X  X
 0  0  1  0  0  0  0  1  0  0  X
 0  0  0  1  0  0  0  1  0  X  X
 0  0  0  0  1  0  0  1  1 X+1 X+1
 0  0  0  0  0  1  0  1 X+1  1 X+1
 0  0  0  0  0  0  1  1 X+1  1  1
 0  0  0  0  0  0  0  X  X  0  X

generates a code of length 11 over Z2[X]/(X^2) who´s minimum homogenous weight is 4.

Homogenous weight enumerator: w(x)=1x^0+164x^4+64x^5+808x^6+896x^7+2726x^8+4032x^9+4472x^10+6400x^11+4556x^12+4032x^13+2648x^14+896x^15+865x^16+64x^17+136x^18+8x^20

The gray image is a linear code over GF(2) with n=22, k=15 and d=4.
As d=4 is an upper bound for linear (22,15,2)-codes, this code is optimal over Z2[X]/(X^2) for dimension 15.
This code was found by Heurico 1.16 in 1.94 seconds.