The generator matrix

 1  0  1  1  1 X^3+X^2+X  X  1  1 X^3+X^2  1  1  1  1 X^2+X  1 X^3  1  1  1 X^2  1  1 X^3+X  1  1  1 X^2 X^2  1  1  1  1  1
 0  1 X+1 X^2+X X^3+X^2+1  1  1 X^3+X^2 X^2+X+1  1 X^3+X^2+X X^2+1  X X^3+1  1 X^3  1 X+1 X^3+X X^3+X^2+X+1  1 X^2  1  1 X^3+X^2+X X^2  1  1  1 X^2+X+1 X^2+X X^3+X^2 X^3+X^2+X+1 X^2+1
 0  0 X^2  0 X^3+X^2 X^2 X^3+X^2  0 X^3+X^2  0 X^2 X^3 X^3+X^2 X^3  0 X^2 X^2 X^3  0 X^3 X^3+X^2 X^2 X^2  0 X^2 X^3+X^2 X^2 X^3+X^2 X^3+X^2 X^3+X^2 X^3  0  0 X^3
 0  0  0 X^3  0  0 X^3  0 X^3 X^3  0  0  0 X^3  0 X^3  0  0 X^3 X^3 X^3  0 X^3 X^3 X^3  0  0  0 X^3 X^3  0 X^3  0 X^3
 0  0  0  0 X^3  0 X^3 X^3  0 X^3 X^3 X^3  0 X^3 X^3  0 X^3  0 X^3  0  0  0  0 X^3 X^3  0  0  0 X^3 X^3 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+190x^30+112x^31+770x^32+400x^33+1172x^34+400x^35+758x^36+112x^37+164x^38+4x^40+4x^42+2x^44+6x^46+1x^48

The gray image is a linear code over GF(2) with n=272, k=12 and d=120.
This code was found by Heurico 1.16 in 86.1 seconds.