The generator matrix

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

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+672x^29+2243x^30+4658x^31+7892x^32+10824x^33+12929x^34+11032x^35+7982x^36+4480x^37+1923x^38+674x^39+165x^40+40x^41+15x^42+4x^43+2x^46

The gray image is a linear code over GF(2) with n=272, k=16 and d=116.
This code was found by Heurico 1.16 in 20.2 seconds.