The generator matrix

 1  0  0  1  1  1 X^3 X^2  1  1  0  1  1  X  1  1 X^3+X^2+X  1  1  0 X^3+X X^3+X^2+X  1  1  1  1 X^3+X^2 X^3+X^2+X X^3+X X^2  X X^3 X^2+X  1
 0  1  0  0 X^3+X^2+1 X^3+X^2+1  1  X X^3 X^2+1  1 X^3 X^2+1  1 X^3+X X^2+X+1  1 X^3+X^2+X X+1  1 X^2+X X^2 X^3+X+1 X^3+X X^2  1  1  1  1  1  1  1  1 X^3+X
 0  0  1 X+1 X^2+X+1 X^2 X^2+X+1  1 X^3+X^2+X X^3+1 X^3+1 X^3+X^2+1 X^3+X  X X^3+X^2+1  1 X^2+X+1  X  0 X^3+X^2+X  1  1 X^3+X X^3+X^2+X+1 X^3+X^2+X  1 X^2+1 X^3+X^2 X^3+X^2 X^3+X^2 X^2+X X^3+X^2+X X^3+X+1 X^2+1
 0  0  0 X^3 X^3  0 X^3 X^3  0 X^3 X^3 X^3  0 X^3  0  0  0 X^3 X^3  0 X^3 X^3 X^3  0 X^3  0  0 X^3  0 X^3  0 X^3 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+155x^30+784x^31+990x^32+1630x^33+1398x^34+1432x^35+811x^36+652x^37+191x^38+88x^39+29x^40+22x^41+8x^42+1x^44

The gray image is a linear code over GF(2) with n=272, k=13 and d=120.
This code was found by Heurico 1.16 in 0.64 seconds.