The generator matrix

 1  0  0  0  1  1  1 X^2+X  1 X^3+X  1 X^2  1
 0  1  0  0  X X^2+1 X+1  1  0 X^2 X^3+X^2+X  X X^2
 0  0  1  0 X+1  1  X  1  0  1 X^3+X^2  1 X^2
 0  0  0  1  1  X  1  1 X^2+X+1 X^2+X X^3+X^2+X+1  X  0
 0  0  0  0 X^2  0  0 X^3 X^3+X^2 X^3+X^2 X^3 X^3+X^2 X^3

generates a code of length 13 over Z2[X]/(X^4) who´s minimum homogenous weight is 8.

Homogenous weight enumerator: w(x)=1x^0+172x^8+1160x^9+5493x^10+20120x^11+60008x^12+88368x^13+59623x^14+20432x^15+5575x^16+968x^17+195x^18+24x^19+4x^20+1x^22

The gray image is a linear code over GF(2) with n=104, k=18 and d=32.
This code was found by Heurico 1.16 in 48.8 seconds.