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

generates a code of length 70 over Z2[X]/(X^4) who´s minimum homogenous weight is 64.

Homogenous weight enumerator: w(x)=1x^0+130x^64+104x^65+250x^66+276x^67+516x^68+500x^69+630x^70+588x^71+433x^72+216x^73+166x^74+84x^75+98x^76+12x^77+52x^78+12x^79+18x^80+4x^82+2x^84+2x^86+1x^88+1x^112

The gray image is a linear code over GF(2) with n=560, k=12 and d=256.
This code was found by Heurico 1.16 in 0.844 seconds.