The generator matrix

 1  0  1  1  1 X^2+X  1  1 X^3+X^2  1  1 X^3+X  1  1  0  1  1 X^2+X  1  1  1  1 X^3+X^2  1  1  1  1  1 X^3+X  1  1  1  1  1  1
 0  1 X+1 X^2+X X^2+1  1 X^3+X^2 X^3+X^2+X+1  1 X^3+X X^3+1  1  0 X+1  1 X^2+X X^2+1  1 X^3+X^2 X^2+X X^3+X^2+X+1 X^2+1  1 X^3+1 X^3+X X^3+X  0  0  1 X^3+1 X^2+1 X+1 X^3+1 X+1  0
 0  0 X^3  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 X^3  0 X^3  0 X^3 X^3  0 X^3  0  0 X^3 X^3  0
 0  0  0 X^3  0  0 X^3 X^3  0 X^3  0 X^3  0 X^3 X^3 X^3  0  0 X^3 X^3  0  0  0  0 X^3  0 X^3  0  0 X^3 X^3  0 X^3  0  0
 0  0  0  0 X^3  0 X^3  0 X^3 X^3 X^3 X^3  0 X^3  0  0 X^3  0  0 X^3 X^3 X^3  0  0  0  0 X^3  0  0  0 X^3 X^3 X^3  0 X^3
 0  0  0  0  0 X^3  0 X^3 X^3 X^3  0 X^3 X^3  0 X^3  0 X^3  0  0 X^3  0 X^3  0  0 X^3 X^3  0 X^3  0 X^3 X^3 X^3 X^3 X^3 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+33x^30+136x^31+159x^32+768x^33+329x^34+1264x^35+319x^36+768x^37+144x^38+136x^39+27x^40+3x^42+5x^44+2x^46+1x^48+1x^54

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 0.125 seconds.