The generator matrix

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

generates a code of length 13 over Z2[X]/(X^3) who´s minimum homogenous weight is 7.

Homogenous weight enumerator: w(x)=1x^0+48x^7+400x^8+1164x^9+3846x^10+7084x^11+13126x^12+14104x^13+13238x^14+7144x^15+3807x^16+1116x^17+386x^18+60x^19+10x^20+2x^22

The gray image is a linear code over GF(2) with n=52, k=16 and d=14.
This code was found by Heurico 1.13 in 1.86 seconds.