The generator matrix

 1  1  1  1  1  1  1  1  1  1  1  1  1  X  1  1  X  1  1  X  1  X  1  X  X  1  X
 0  X  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  X  X  X  X  X  X  X
 0  0  X  0  0  0  0  0  0  0  0  0  0  0  0  0  0  X  X  X  X  X  X  0  0  X  0
 0  0  0  X  0  0  0  0  0  0  0  0  0  0  0  0  X  0  X  X  X  0  0  0  X  X  X
 0  0  0  0  X  0  0  0  0  0  0  0  0  0  X  X  X  X  0  X  0  0  0  0  X  X  0
 0  0  0  0  0  X  0  0  0  0  0  0  0  X  X  X  0  0  0  X  X  0  X  0  X  0  0
 0  0  0  0  0  0  X  0  0  0  0  0  0  X  0  X  0  X  0  0  0  0  X  X  X  X  0
 0  0  0  0  0  0  0  X  0  0  0  0  0  X  X  0  X  X  X  X  0  X  X  X  X  X  0
 0  0  0  0  0  0  0  0  X  0  0  0  X  X  X  X  X  X  0  0  X  0  0  X  0  0  X
 0  0  0  0  0  0  0  0  0  X  0  0  X  0  0  X  X  X  X  0  0  X  0  X  X  0  X
 0  0  0  0  0  0  0  0  0  0  X  0  X  0  X  X  0  0  X  0  X  0  0  X  X  X  X
 0  0  0  0  0  0  0  0  0  0  0  X  X  X  0  0  0  X  0  X  0  X  0  0  X  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+216x^16+462x^20+224x^22+1694x^24+1120x^26+2438x^28+672x^30+983x^32+32x^34+298x^36+50x^40+2x^44

The gray image is a linear code over GF(2) with n=54, k=13 and d=16.
This code was found by Heurico 1.16 in 4.33 seconds.