The generator matrix

 1  0  1  1  1 X^2+X  1  1 X^3+X^2  1  1 X^3+X  1  1  0  1  1 X^2+X  1  1  1 X^3+X  1  1 X^3+X^2  1  0  1 X^3+X^2  1  1  1  1  1
 0  1 X+1 X^2+X X^2+1  1 X^3+X^2 X^3+X^2+X+1  1 X^3+X X^3+1  1  0 X+1  1 X^2+X X^2+1  1 X^3+X^2 X^3+X^2+X+1 X^3+X  1  0 X^3+1  1 X^3+1  1 X^2+1  1 X+1 X^2+1 X^3+X^2 X+1 X^3+1
 0  0 X^3  0  0  0  0 X^3  0 X^3 X^3 X^3  0  0  0 X^3 X^3 X^3  0  0  0  0  0 X^3  0 X^3 X^3  0  0 X^3 X^3 X^3  0 X^3
 0  0  0 X^3  0  0 X^3 X^3  0 X^3  0 X^3  0 X^3 X^3 X^3  0  0 X^3  0 X^3 X^3 X^3 X^3  0  0  0  0  0  0 X^3  0  0 X^3
 0  0  0  0 X^3  0 X^3  0 X^3 X^3 X^3 X^3  0 X^3  0  0 X^3  0  0 X^3 X^3  0  0 X^3  0  0 X^3 X^3 X^3 X^3  0 X^3  0  0
 0  0  0  0  0 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 X^3 X^3  0  0 X^3  0 X^3 X^3 X^3  0  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+182x^30+128x^31+773x^32+384x^33+1184x^34+384x^35+752x^36+128x^37+164x^38+8x^40+6x^46+2x^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 1.86 seconds.