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

generates a code of length 65 over Z2[X]/(X^4) who´s minimum homogenous weight is 60.

Homogenous weight enumerator: w(x)=1x^0+291x^60+48x^61+524x^62+280x^63+809x^64+376x^65+758x^66+264x^67+350x^68+56x^69+208x^70+98x^72+14x^74+18x^76+1x^108

The gray image is a linear code over GF(2) with n=520, k=12 and d=240.
This code was found by Heurico 1.16 in 2.62 seconds.