The generator matrix

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

generates a code of length 27 over Z2[X]/(X^3) who´s minimum homogenous weight is 16.

Homogenous weight enumerator: w(x)=1x^0+170x^16+80x^18+24x^19+408x^20+256x^21+320x^22+224x^23+1928x^24+1216x^25+3448x^26+528x^27+3024x^28+1408x^29+1696x^30+224x^31+525x^32+192x^33+480x^34+24x^35+88x^36+96x^38+24x^42

The gray image is a linear code over GF(2) with n=108, k=14 and d=32.
This code was found by Heurico 1.16 in 3.5 seconds.