The generator matrix

 1  1  1  1  1  1  1  1  1  1  1  1  1  1  X  0  1  1  X  0  X  X X^2  X  1  1  1  X  1  X  1  1 X^3  X  X
 0  X  0 X^3+X^2+X X^2 X^2+X X^3+X^2  X X^3  0 X^2+X X^2+X X^2 X^2 X^2+X  X  X X^3+X X^3+X  X X^3+X^2+X  X  X X^3+X^2+X X^3 X^2  X X^3+X^2+X X^3+X^2+X X^2 X^3 X^2+X X^2  X X^3+X^2+X
 0  0 X^3+X^2  0 X^2  0  0 X^3  0 X^2 X^2 X^2 X^2 X^3 X^3+X^2 X^3 X^2 X^3+X^2 X^2 X^3  0 X^3  0 X^3+X^2 X^3+X^2 X^3+X^2 X^3+X^2 X^2 X^3+X^2 X^3 X^3  0 X^3+X^2 X^3+X^2 X^3
 0  0  0 X^3+X^2  0  0 X^3 X^2 X^2 X^2 X^2 X^3 X^3+X^2 X^2 X^3+X^2  0 X^2  0 X^3+X^2 X^3 X^2 X^3+X^2 X^3+X^2  0 X^3 X^3 X^3  0  0 X^3+X^2 X^2 X^3  0  0 X^2
 0  0  0  0 X^3 X^3 X^3 X^3  0  0  0 X^3 X^3 X^3  0  0 X^3  0  0  0 X^3 X^3  0 X^3 X^3  0 X^3 X^3  0  0  0  0  0  0 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+115x^30+156x^31+362x^32+420x^33+717x^34+632x^35+684x^36+440x^37+286x^38+140x^39+85x^40+4x^41+29x^42+18x^44+5x^46+2x^48

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 0.219 seconds.