The generator matrix

 1  0  1  1  1  0  1  1  1  1  X
 0  1  1  0  1  1  0  0  0  0  0
 0  0  X  0  0  0  0  0  0  0  X
 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

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+63x^4+56x^6+754x^8+1048x^10+1510x^12+424x^14+221x^16+8x^18+11x^20

The gray image is a linear code over GF(2) with n=22, k=12 and d=4.
This code was found by Heurico 1.16 in 0.0354 seconds.