The generator matrix

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

generates a code of length 13 over Z2[X]/(X^3) who´s minimum homogenous weight is 4.

Homogenous weight enumerator: w(x)=1x^0+69x^4+144x^6+580x^8+1016x^10+1536x^11+2370x^12+5120x^13+2040x^14+1536x^15+1313x^16+360x^18+265x^20+24x^22+10x^24

The gray image is a linear code over GF(2) with n=52, k=14 and d=8.
This code was found by Heurico 1.16 in 0.249 seconds.