The generator matrix

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

generates a code of length 34 over Z2[X]/(X^4) who´s minimum homogenous weight is 27.

Homogenous weight enumerator: w(x)=1x^0+84x^27+874x^28+2988x^29+7933x^30+16646x^31+31043x^32+44956x^33+52619x^34+45206x^35+31704x^36+16828x^37+7533x^38+2562x^39+844x^40+244x^41+41x^42+14x^43+14x^44+8x^45+2x^46

The gray image is a linear code over GF(2) with n=272, k=18 and d=108.
This code was found by Heurico 1.16 in 229 seconds.