The generator matrix

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

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+234x^31+292x^32+368x^33+353x^34+340x^35+200x^36+168x^37+46x^38+34x^39+1x^40+8x^41+1x^42+2x^44

The gray image is a linear code over GF(2) with n=272, k=11 and d=124.
This code was found by Heurico 1.16 in 135 seconds.