The generator matrix

 1  0  1  1  1  0  1  1  1  1  1
 0  1  1  0  1  1  0  0  0  0  0
 0  0  X  0  0  0  0  0  0  X  0
 0  0  0  X  0  0  0  0  0  X  0
 0  0  0  0  X  0  0  0  0  X  0
 0  0  0  0  0  X  0  0  0  X  0
 0  0  0  0  0  0  X  0  0  X  0
 0  0  0  0  0  0  0  X  0  X  0
 0  0  0  0  0  0  0  0  X  X  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+44x^4+49x^6+214x^8+256x^9+204x^10+512x^11+204x^12+256x^13+214x^14+49x^16+44x^18+1x^22

The gray image is a linear code over GF(2) with n=22, k=11 and d=4.
This code was found by Heurico 1.16 in 0.0117 seconds.