The generator matrix

 1  0  0  0  1  1  1  1  0  1  X  1  1  X  X  0  1  0  X  0  0  1  1  1  1  1  1
 0  1  0  0  1  0  0  1  1  0  0  1  1  1  1  1  X  1  1  X  X  1 X+1 X+1  1  0  0
 0  0  1  0  1  0  1  0  1  1  1  1  0  0  1  1  0  1  1  X  1 X+1  1  X  0  1  0
 0  0  0  1  1  1  0  0  1  1 X+1  0 X+1  1  0  1  X  1  X  1  1 X+1  X  1  1  1  0
 0  0  0  0  X  0  0  0  0  0  X  0  X  0  X  0  0  0  X  X  0  0  X  X  X  0  0
 0  0  0  0  0  X  0  0  0  0  0  0  X  X  X  0  X  X  X  0  0  X  X  X  X  0  0
 0  0  0  0  0  0  X  0  0  0  0  X  0  X  X  0  0  X  0  X  X  0  X  0  X  X  0
 0  0  0  0  0  0  0  X  0  0  X  X  0  0  X  0  0  X  X  0  X  X  X  X  X  X  0
 0  0  0  0  0  0  0  0  X  0  X  X  X  0  0  X  0  0  X  X  X  X  X  0  0  0  0
 0  0  0  0  0  0  0  0  0  X  X  0  X  X  0  X  X  X  0  0  0  0  X  0  0  0  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+26x^16+38x^17+153x^18+200x^19+345x^20+466x^21+688x^22+996x^23+1267x^24+1540x^25+1639x^26+1688x^27+1570x^28+1548x^29+1258x^30+1040x^31+763x^32+470x^33+319x^34+160x^35+117x^36+34x^37+36x^38+12x^39+7x^40+1x^42+2x^46

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 12.8 seconds.