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  1  1  1  1  1  1  1  1 X^3+X^2  1  1  X  1  1  1 X^2  1  1  X  1  X  1 X^3+X^2  1 X^3+X^2  1  0  X  X  1  1  1 X^2  0  X  1  1  1  1  X
 0  X  0  X X^3  0 X^2+X X^3+X^2+X  0 X^3 X^3+X  X X^3 X^3+X^2+X X^3 X^3+X^2+X X^3 X^2+X X^2+X X^2 X^3+X X^3+X^2 X^3+X^2 X^2+X  X X^2 X^3+X^2  X X^3+X^2+X  0  X  0 X^3+X^2 X^2+X X^3+X^2 X^3+X  X X^3 X^2  0 X^3+X X^3+X X^2+X X^3+X^2 X^2 X^3+X^2 X^3+X X^2 X^2+X X^3+X^2+X  X X^3+X  X X^2+X  X X^3+X^2  0 X^3 X^3+X  X X^2+X  0  X  X X^3+X^2+X X^2+X X^3+X X^3+X^2 X^2 X^2+X
 0  0  X  X  0 X^3+X^2+X X^2+X X^3 X^2 X^3+X^2+X X^3+X^2+X X^2 X^3+X X^2 X^3+X^2  X X^3 X^3 X^2+X  X X^2 X^2+X X^3 X^2+X X^2+X X^3+X  0 X^2 X^3 X^3 X^3+X  X X^3 X^3+X  X X^3 X^3+X X^2  X  X X^3+X^2 X^2 X^3+X^2 X^2 X^2+X  X X^2+X X^3+X^2 X^3+X X^3  0 X^3+X X^3 X^2 X^3+X^2+X X^3+X^2+X  X X^3+X X^2+X X^3+X^2+X X^3+X X^3+X^2+X X^3  X X^2+X  X X^2 X^2+X  X X^3
 0  0  0 X^2 X^2 X^3+X^2  0 X^3+X^2 X^2 X^3 X^3+X^2  0 X^2 X^3+X^2  0 X^3 X^3  0 X^3  0 X^2 X^3+X^2 X^2 X^3+X^2  0 X^3  0 X^3+X^2 X^3 X^3+X^2 X^3+X^2 X^3+X^2 X^3+X^2 X^2 X^3+X^2 X^2 X^3 X^3 X^3+X^2 X^3 X^3 X^2 X^2  0 X^2  0 X^2 X^2 X^2 X^2 X^3+X^2  0  0 X^3  0  0 X^3 X^3+X^2 X^3 X^3+X^2 X^3  0 X^3 X^3+X^2 X^3 X^3+X^2  0 X^2 X^3+X^2  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+188x^65+233x^66+362x^67+637x^68+468x^69+594x^70+494x^71+452x^72+198x^73+149x^74+90x^75+73x^76+76x^77+32x^78+30x^79+4x^80+14x^81+1x^112

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 32.9 seconds.