The generator matrix

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

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+298x^29+1082x^30+2492x^31+3601x^32+5812x^33+6176x^34+5962x^35+3852x^36+2106x^37+810x^38+440x^39+76x^40+40x^41+12x^42+2x^43+4x^44+2x^48

The gray image is a linear code over GF(2) with n=272, k=15 and d=116.
This code was found by Heurico 1.16 in 6.64 seconds.