The generator matrix

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

generates a code of length 13 over Z2[X]/(X^2) who´s minimum homogenous weight is 4.

Homogenous weight enumerator: w(x)=1x^0+53x^4+48x^6+340x^8+424x^10+1266x^12+936x^14+721x^16+120x^18+169x^20+8x^22+10x^24

The gray image is a linear code over GF(2) with n=26, k=12 and d=4.
This code was found by Heurico 1.16 in 0.0366 seconds.