The generator matrix

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

generates a code of length 34 over Z2[X]/(X^4) who´s minimum homogenous weight is 31.

Homogenous weight enumerator: w(x)=1x^0+304x^31+842x^32+600x^33+905x^34+468x^35+527x^36+220x^37+142x^38+60x^39+13x^40+12x^41+1x^42+1x^44

The gray image is a linear code over GF(2) with n=272, k=12 and d=124.
This code was found by Heurico 1.16 in 2.14 seconds.