The generator matrix

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

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+40x^29+101x^30+146x^31+221x^32+342x^33+394x^34+332x^35+217x^36+120x^37+70x^38+26x^39+6x^40+6x^41+10x^42+8x^43+2x^44+4x^45+1x^46+1x^52

The gray image is a linear code over GF(2) with n=272, k=11 and d=116.
This code was found by Heurico 1.16 in 0.079 seconds.