The generator matrix

 1  1  1  1  1  1  1  1  1  1  X
 0  X  0  0  0  0  0  0  0  0  X
 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+55x^4+330x^8+512x^10+974x^12+165x^16+11x^20

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 0.0108 seconds.