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

generates a code of length 70 over Z2[X]/(X^4) who´s minimum homogenous weight is 65.

Homogenous weight enumerator: w(x)=1x^0+214x^65+293x^66+324x^67+406x^68+634x^69+576x^70+530x^71+385x^72+262x^73+187x^74+124x^75+40x^76+38x^77+30x^78+42x^79+4x^81+1x^82+4x^83+1x^114

The gray image is a linear code over GF(2) with n=560, k=12 and d=260.
This code was found by Heurico 1.16 in 83.5 seconds.