The generator matrix

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

generates a code of length 34 over Z2[X]/(X^4) who´s minimum homogenous weight is 30.

Homogenous weight enumerator: w(x)=1x^0+93x^30+128x^31+261x^32+320x^33+498x^34+336x^35+192x^36+96x^37+73x^38+16x^39+20x^40+8x^42+5x^44+1x^52

The gray image is a linear code over GF(2) with n=272, k=11 and d=120.
This code was found by Heurico 1.16 in 0.078 seconds.