The generator matrix

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

generates a code of length 27 over Z2[X]/(X^4) who´s minimum homogenous weight is 20.

Homogenous weight enumerator: w(x)=1x^0+29x^20+36x^22+95x^24+362x^26+1024x^27+338x^28+98x^30+39x^32+14x^34+7x^36+2x^38+1x^40+2x^44

The gray image is a linear code over GF(2) with n=216, k=11 and d=80.
This code was found by Heurico 1.16 in 0.062 seconds.