The generator matrix

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

generates a code of length 27 over Z2[X]/(X^3) who´s minimum homogenous weight is 22.

Homogenous weight enumerator: w(x)=1x^0+26x^22+58x^23+111x^24+140x^25+144x^26+154x^27+112x^28+96x^29+65x^30+38x^31+30x^32+20x^33+18x^34+6x^35+2x^36+3x^38

The gray image is a linear code over GF(2) with n=108, k=10 and d=44.
This code was found by Heurico 1.16 in 0.0313 seconds.