The generator matrix

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

generates a code of length 27 over Z2[X]/(X^4) who´s minimum homogenous weight is 22.

Homogenous weight enumerator: w(x)=1x^0+262x^22+1610x^23+3806x^24+7968x^25+11655x^26+14526x^27+12197x^28+8280x^29+3476x^30+1326x^31+327x^32+72x^33+15x^34+10x^35+5x^36

The gray image is a linear code over GF(2) with n=216, k=16 and d=88.
This code was found by Heurico 1.16 in 14.8 seconds.