The generator matrix

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

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+112x^4+168x^5+600x^6+1208x^7+2630x^8+3912x^9+4840x^10+5784x^11+4900x^12+3960x^13+2536x^14+1192x^15+673x^16+152x^17+88x^18+8x^19+4x^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 1.11 in 0.0577 seconds.