The generator matrix

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

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+72x^8+180x^9+306x^10+1140x^11+1154x^12+2472x^13+1166x^14+1160x^15+303x^16+164x^17+62x^18+4x^19+6x^20+2x^22

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 0.202 seconds.