The generator matrix

 1  0  0  1  1  1  0  1  1  1  1  X
 0  1  0  1  0  1  1  0  0  0  0  0
 0  0  1  1  1  0  1  0  0  0  0  0
 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+90x^4+192x^6+1679x^8+3392x^10+5676x^12+3392x^14+1679x^16+192x^18+90x^20+1x^24

The gray image is a linear code over GF(2) with n=24, k=14 and d=4.
This code was found by Heurico 1.16 in 0.483 seconds.