The generator matrix

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

generates a code of length 34 over Z2[X]/(X^4) who´s minimum homogenous weight is 30.

Homogenous weight enumerator: w(x)=1x^0+96x^30+128x^31+399x^32+192x^33+456x^34+192x^35+356x^36+128x^37+68x^38+17x^40+12x^42+2x^44+1x^48

The gray image is a linear code over GF(2) with n=272, k=11 and d=120.
This code was found by Heurico 1.16 in 0.062 seconds.