The generator matrix

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

generates a code of length 12 over Z2[X]/(X^2) who´s minimum homogenous weight is 4.

Homogenous weight enumerator: w(x)=1x^0+66x^4+495x^8+2972x^12+495x^16+66x^20+1x^24

The gray image is a linear code over GF(2) with n=24, k=12 and d=4.
This code was found by Heurico 1.16 in 0.0499 seconds.