The generator matrix

 1  1  1  1  1  1  1  X  1  X  X  1  1
 0  X  0  0  0  X  0  0  X  X  X X^2+X X^2
 0  0  X  0  0  X  0  X  X X^2  X  X  0
 0  0  0  X  0  X  X X^2+X  0 X^2+X  X  X  X
 0  0  0  0  X  X  X X^2 X^2  0  X  0 X^2
 0  0  0  0  0 X^2  0 X^2 X^2 X^2  0 X^2 X^2
 0  0  0  0  0  0 X^2  0 X^2  0  0 X^2 X^2

generates a code of length 13 over Z2[X]/(X^3) who´s minimum homogenous weight is 8.

Homogenous weight enumerator: w(x)=1x^0+311x^8+480x^10+384x^11+2416x^12+1280x^13+1920x^14+384x^15+823x^16+160x^18+32x^20+1x^24

The gray image is a linear code over GF(2) with n=52, k=13 and d=16.
This code was found by Heurico 1.16 in 13.5 seconds.