The generator matrix

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

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+78x^30+142x^31+224x^32+468x^33+246x^34+464x^35+187x^36+140x^37+91x^38+2x^39+1x^40+2x^44+1x^46+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 0.047 seconds.