The generator matrix

 1  0  1  1 X^2  1  1  1 X^2+X  1  1  0 X^3+X  1  1  1  1 X^2 X^3  1  1  1  1  1 X^3  1 X^3+X  1  1  1  1 X^3+X^2+X  1 X^2+X  1  1  1 X^2+X  1 X^3+X  1  1  1  1  1 X^3+X^2+X  0 X^3+X^2+X  0  1  1  X  1  1 X^3+X^2+X  1  1  1 X^2  1  1  1  0  0 X^2  1  1  1  X  1
 0  1  1 X^2+X  1 X^2+X+1 X^2 X^3+1  1 X+1 X^3+X^2+X  1  1  0 X^3+X^2+1 X^3 X^3+1  1  1 X^2+X X+1  X X+1 X^3+X^2  1  1  1 X^2 X^3 X^2+1 X^3+1  1 X^3+X^2+X+1  1 X^3+X X^2+X+1 X^3  1 X^3+X+1  1 X+1  X X^2+1  X X^3+X^2+X  1  1  1  1 X^3+X^2+X+1 X^3+X^2+X X^3+X^2+X  1  X  1 X^3 X^2+X  X  1 X^3+X^2+1 X^3+X^2  X  X  X  1 X^2+X+1 X^3+X  1 X^3+X X^2+1
 0  0  X  0 X^3+X  X X^3+X X^3  0 X^3 X^3+X X^3+X^2+X X^2 X^3+X^2 X^3+X^2 X^3+X^2+X X^3+X^2+X X^2+X X^3+X X^2+X X^2 X^2 X^2+X X^2+X X^3+X^2 X^2 X^3+X^2+X X^2 X^3+X X^3  X X^3+X^2 X^3+X^2 X^3+X^2+X X^3+X^2+X  X  0  X X^3+X^2  0 X^3+X^2+X X^3+X^2 X^3+X^2+X X^3+X X^3 X^3+X X^3+X^2 X^3  0  0 X^2 X^2+X X^3 X^3+X^2+X X^3+X^2 X^3+X^2+X X^3+X X^2+X X^3+X^2 X^2 X^3+X^2+X  X X^3+X X^3+X  0  0 X^2 X^2+X  X X^2+X
 0  0  0 X^3  0 X^3 X^3 X^3 X^3  0  0 X^3 X^3  0 X^3 X^3  0  0 X^3  0  0 X^3 X^3  0  0  0 X^3 X^3  0  0 X^3  0  0  0  0  0 X^3 X^3 X^3 X^3  0  0 X^3 X^3  0  0 X^3  0 X^3 X^3  0  0  0 X^3 X^3  0 X^3 X^3  0  0 X^3  0  0 X^3 X^3  0  0 X^3 X^3  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+482x^66+128x^67+1085x^68+160x^69+680x^70+96x^71+846x^72+96x^73+338x^74+32x^75+80x^76+60x^78+8x^82+2x^84+1x^88+1x^92

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 0.532 seconds.