The generator matrix

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

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+63x^4+67x^6+506x^8+501x^10+1024x^11+910x^12+2048x^13+910x^14+1024x^15+501x^16+506x^18+67x^20+63x^22+1x^26

The gray image is a linear code over GF(2) with n=26, k=13 and d=4.
This code was found by Heurico 1.16 in 0.146 seconds.