The generator matrix

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

generates a code of length 34 over Z2[X]/(X^4) who´s minimum homogenous weight is 28.

Homogenous weight enumerator: w(x)=1x^0+257x^28+1504x^29+4198x^30+9096x^31+14404x^32+23756x^33+23948x^34+24688x^35+14787x^36+8984x^37+3550x^38+1256x^39+511x^40+92x^41+32x^42+8x^44

The gray image is a linear code over GF(2) with n=272, k=17 and d=112.
This code was found by Heurico 1.16 in 62.9 seconds.