The generator matrix

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

generates a code of length 34 over Z2[X]/(X^4) who´s minimum homogenous weight is 29.

Homogenous weight enumerator: w(x)=1x^0+92x^29+163x^30+370x^31+265x^32+1066x^33+221x^34+1110x^35+210x^36+362x^37+129x^38+46x^39+14x^40+14x^41+11x^42+10x^43+6x^44+2x^45+4x^46

The gray image is a linear code over GF(2) with n=272, k=12 and d=116.
This code was found by Heurico 1.16 in 0.578 seconds.