The generator matrix

 1  0  1  1  1  0  1  1  0  1  1  0  1  1  1  0  1  0  1  0  X  0  0  X  0  0  1
 0  1  1  0  1  1  0  1  1  0  1  1  0  X X+1  1  1  1  0  1  0  1  1  0  1  X  0
 0  0  X  0  0  0  0  0  0  0  0  0  0  0  0  0  X  X  0  X  X  0  0  X  0  X  X
 0  0  0  X  0  0  0  0  0  0  0  0  0  X  0  0  0  X  X  0  X  0  X  0  X  X  0
 0  0  0  0  X  0  0  0  0  0  0  0  0  0  0  X  X  0  X  0  0  X  X  X  0  0  X
 0  0  0  0  0  X  0  0  0  0  0  0  0  0  0  0  0  0  0  0  X  X  X  X  X  X  X
 0  0  0  0  0  0  X  0  0  0  0  0  0  X  X  X  X  0  0  X  X  0  X  0  0  X  X
 0  0  0  0  0  0  0  X  0  0  0  0  0  0  X  0  0  0  X  X  X  X  0  0  X  0  X
 0  0  0  0  0  0  0  0  X  0  0  0  X  X  0  X  0  0  X  X  0  0  0  X  0  X  0
 0  0  0  0  0  0  0  0  0  X  0  0  X  0  0  0  0  X  X  0  0  0  X  X  X  0  X
 0  0  0  0  0  0  0  0  0  0  X  0  X  X  0  X  X  0  0  X  X  X  X  X  0  0  0
 0  0  0  0  0  0  0  0  0  0  0  X  X  X  X  0  X  X  X  X  0  X  X  X  0  X  0

generates a code of length 27 over Z2[X]/(X^2) who´s minimum homogenous weight is 16.

Homogenous weight enumerator: w(x)=1x^0+193x^16+70x^18+846x^20+1002x^22+2668x^24+3100x^26+4220x^28+1780x^30+1785x^32+190x^34+470x^36+2x^38+56x^40+1x^48

The gray image is a linear code over GF(2) with n=54, k=14 and d=16.
This code was found by Heurico 1.16 in 21.4 seconds.