The generator matrix

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

generates a code of length 14 over Z2[X]/(X^2) who´s minimum homogenous weight is 4.

Homogenous weight enumerator: w(x)=1x^0+66x^4+12x^6+495x^8+220x^10+924x^12+2048x^13+792x^14+2048x^15+495x^16+792x^18+66x^20+220x^22+1x^24+12x^26

The gray image is a linear code over GF(2) with n=28, k=13 and d=4.
This code was found by Heurico 1.16 in 0.797 seconds.