The generator matrix

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

generates a code of length 70 over Z2[X]/(X^4) who´s minimum homogenous weight is 66.

Homogenous weight enumerator: w(x)=1x^0+124x^66+690x^67+658x^68+586x^69+545x^70+424x^71+314x^72+286x^73+156x^74+114x^75+65x^76+84x^77+18x^78+24x^79+1x^80+4x^82+1x^88+1x^90

The gray image is a linear code over GF(2) with n=560, k=12 and d=264.
This code was found by Heurico 1.16 in 2.5 seconds.