The generator matrix

 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  X  X  1  X  1  X  X  X  1  1
 0 X^3  0  0  0  0  0  0  0  0 X^3 X^3  0  0  0 X^3  0  0  0  0 X^3 X^3 X^3 X^3 X^3 X^3  0 X^3  0 X^3  0 X^3  0 X^3
 0  0 X^3  0  0  0  0  0  0 X^3 X^3 X^3  0 X^3 X^3  0  0  0 X^3 X^3 X^3 X^3 X^3 X^3 X^3  0  0  0  0  0 X^3  0 X^3 X^3
 0  0  0 X^3  0  0  0  0  0 X^3  0 X^3 X^3 X^3  0 X^3  0 X^3 X^3  0  0  0  0  0  0  0 X^3  0 X^3 X^3 X^3  0 X^3 X^3
 0  0  0  0 X^3  0  0  0 X^3  0  0 X^3 X^3  0 X^3 X^3  0  0 X^3 X^3  0  0  0 X^3 X^3  0  0 X^3 X^3 X^3  0 X^3 X^3 X^3
 0  0  0  0  0 X^3  0  0 X^3  0 X^3  0  0 X^3 X^3 X^3  0 X^3 X^3  0  0  0 X^3 X^3  0  0 X^3  0 X^3  0  0 X^3 X^3  0
 0  0  0  0  0  0 X^3  0 X^3 X^3 X^3  0  0  0 X^3 X^3 X^3  0  0 X^3  0  0  0  0 X^3 X^3 X^3  0 X^3 X^3  0  0 X^3  0
 0  0  0  0  0  0  0 X^3 X^3 X^3  0  0 X^3 X^3 X^3  0 X^3  0 X^3  0  0 X^3 X^3  0 X^3  0  0 X^3  0  0  0 X^3  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+79x^28+24x^30+123x^32+1616x^34+110x^36+24x^38+49x^40+19x^44+2x^48+1x^56

The gray image is a linear code over GF(2) with n=272, k=11 and d=112.
This code was found by Heurico 1.16 in 11.7 seconds.