The generator matrix

 1  0  0  0  1  1  1 X^2+X X^2  1  1  1  1
 0  1  0  0  0  1  1  1  X X^2+X  0 X^2+1 X^2+1
 0  0  1  0  1  1  0 X+1  1 X^2+X X^2+X X^2+1  0
 0  0  0  1  1  0 X+1  1  1 X+1 X^2 X^2+X+1 X^2+X+1
 0  0  0  0  X  0 X^2+X  X X^2+X X^2  X  X X^2
 0  0  0  0  0 X^2 X^2  0 X^2 X^2 X^2  0 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+259x^8+504x^9+1980x^10+3600x^11+6419x^12+7152x^13+6538x^14+3616x^15+1980x^16+472x^17+216x^18+16x^19+13x^20+2x^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 1.16 in 2.47 seconds.