The generator matrix

 1  0  0  0  0  1  1  1  X  1  1  X  1
 0  1  0  0  0 X^2 X^2  X  1  X  1  1  0
 0  0  1  0  0 X^2+X  1 X+1 X+1 X^2+1 X^2  1 X^2
 0  0  0  1  0 X+1  1 X^2  1 X^2+1  X X^2+1 X^2
 0  0  0  0  1  1  X X^2+1 X+1 X+1 X^2 X^2+1 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+294x^8+624x^9+1521x^10+4184x^11+5382x^12+8768x^13+5263x^14+4304x^15+1629x^16+528x^17+223x^18+24x^19+22x^20+1x^22

The gray image is a linear code over GF(2) with n=52, k=15 and d=16.
This code was found by Heurico 1.13 in 0.985 seconds.