The generator matrix

 1  0  0  0  0  0  1  1  1  1  1  1  1
 0  1  0  0  0  0  0  1  0 X+1  X  X  X
 0  0  1  0  0  0  0  1  1  1  1 X+1  0
 0  0  0  1  0  0  1  0  1  0  0  X  0
 0  0  0  0  1  0  1  0  0  1  1  0  X
 0  0  0  0  0  1  1  1  0  0  1  X X+1
 0  0  0  0  0  0  X  0  0  0  X  X  X
 0  0  0  0  0  0  0  X  0  X  X  0  X
 0  0  0  0  0  0  0  0  X  X  0  0  X

generates a code of length 13 over Z2[X]/(X^2) who´s minimum homogenous weight is 5.

Homogenous weight enumerator: w(x)=1x^0+34x^5+130x^6+314x^7+751x^8+1536x^9+2576x^10+3792x^11+4744x^12+5044x^13+4708x^14+3764x^15+2591x^16+1552x^17+768x^18+320x^19+104x^20+26x^21+10x^22+2x^23+1x^24

The gray image is a linear code over GF(2) with n=26, k=15 and d=5.
This code was found by Heurico 1.16 in 3.64 seconds.