The generator matrix

 1  0  1  1  1 X^3+X^2+X  1  X  1 X^3  1  1 X^2  1  1  1 X^2+X  1  1 X^3+X^2  1 X^3+X  1  1  1  1  1 X^3+X  1  0  1  1  1  1  1 X^3 X^3+X^2+X  1  1  1  1  1 X^3  X X^2  1  1 X^2+X  1  1  X  1  1  1  1  1  1  1  1  1  1  1  1  0  X  1 X^3+X^2+X X^3  1  1
 0  1 X+1 X^2+X X^3+X^2+1  1 X^3+X^2  1 X^2+X+1  1 X^3+X  1  1 X^3 X+1 X^3+X^2+X  1 X^3+X^2+X+1 X^2  1  X  1 X+1 X^3+X^2+X+1 X^2+1 X^3+1  0  1 X^2+1  1 X^3+X^2+X X^3+X+1 X^2 X^3+X^2+1 X^2+X  1  1 X^3+1 X^3+X^2 X^3+X X^2+X X^3+X+1  1  1  1 X^3  1  1  0  X X^2 X^3+X X^3 X^2 X^3+X  X X^3+X^2 X^2 X^3 X^2+X X^2+X X^3+X^2+X X^3  X  1 X^3+X+1  1  X X^3+X^2+X X^3+X^2+1
 0  0 X^2  0 X^3+X^2 X^2  0 X^2 X^3+X^2 X^3+X^2  0 X^2 X^3+X^2 X^2 X^3 X^3+X^2  0 X^3 X^2  0 X^3+X^2  0 X^3 X^3  0 X^3  0  0 X^3  0 X^3+X^2 X^2 X^2 X^3+X^2 X^3 X^2 X^2  0  0 X^3 X^2 X^2 X^3 X^2 X^2 X^3+X^2 X^3+X^2 X^3 X^3+X^2 X^3+X^2 X^3 X^2 X^2  0  0  0 X^3 X^3  0 X^2 X^3 X^3 X^2 X^2  0 X^3+X^2 X^3 X^2  0 X^3+X^2
 0  0  0 X^3  0  0  0  0 X^3 X^3 X^3 X^3 X^3  0 X^3 X^3 X^3  0 X^3 X^3  0  0 X^3  0  0 X^3 X^3  0 X^3  0 X^3  0 X^3  0  0  0 X^3  0 X^3  0  0 X^3 X^3 X^3  0  0 X^3 X^3 X^3 X^3  0  0  0 X^3  0 X^3 X^3  0  0 X^3 X^3  0 X^3 X^3 X^3 X^3 X^3  0  0 X^3
 0  0  0  0 X^3  0 X^3 X^3  0  0 X^3 X^3 X^3 X^3  0 X^3  0  0  0 X^3  0 X^3 X^3 X^3  0 X^3 X^3  0  0 X^3  0 X^3 X^3  0 X^3  0 X^3 X^3  0  0 X^3 X^3  0  0 X^3  0  0 X^3 X^3  0 X^3 X^3  0 X^3  0  0 X^3  0 X^3 X^3  0  0  0 X^3  0  0  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+114x^65+279x^66+494x^67+345x^68+708x^69+335x^70+660x^71+321x^72+462x^73+205x^74+94x^75+33x^76+20x^77+5x^78+2x^80+4x^81+6x^82+2x^84+4x^85+2x^98

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 0.547 seconds.