The generator matrix

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

generates a code of length 35 over Z2[X]/(X^4) who´s minimum homogenous weight is 30.

Homogenous weight enumerator: w(x)=1x^0+40x^30+190x^31+299x^32+548x^33+568x^34+938x^35+591x^36+410x^37+206x^38+154x^39+66x^40+48x^41+17x^42+14x^43+3x^44+2x^45+1x^50

The gray image is a linear code over GF(2) with n=280, k=12 and d=120.
This code was found by Heurico 1.16 in 0.156 seconds.