The generator matrix

 1  0  1  1  1 X^2+X  1  1 X^3+X^2  1  1 X^3+X  1  1  0  1  1 X^2+X  1  1 X^3+X^2  1  1 X^3+X  1  1  0  1  1 X^2+X  1  1 X^3+X^2  1 X^3+X  1  1  1  1  1  1  1  1  1  0 X^2+X X^3+X^2 X^3+X^2+X  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 X^3+X  X  1  1  1  1
 0  1 X+1 X^2+X X^2+1  1 X^3+X^2+X+1 X^3+X^2  1 X^3+X X^3+1  1  0 X+1  1 X^2+X X^2+1  1 X^3+X^2 X^3+X^2+X+1  1 X^3+X X^3+1  1  0 X+1  1 X^2+X X^2+1  1 X^3+X^2 X^3+X^2+X+1  1 X^3+X  1 X^3+1 X^3+X^2+1 X+1  1 X^3+X^2+X+1  0 X^2+X X^3+X^2 X^3+X  1  1  1  1 X^2+1 X^3+X+1 X^2+X+1 X^3+1 X^2+1 X+1 X^3+1 X^3+X^2+X+1 X+1 X^3+X+1 X^2+1 X^3+X^2+1 X^3+X^2+X+1 X^2+X+1 X^3+1  1  1  1 X^3 X^2  0 X^3
 0  0 X^3  0  0  0  0 X^3 X^3 X^3 X^3 X^3  0  0  0  0  0  0 X^3 X^3 X^3 X^3 X^3 X^3  0  0  0 X^3 X^3 X^3  0 X^3  0 X^3 X^3  0 X^3  0  0 X^3 X^3  0 X^3  0 X^3  0 X^3  0  0 X^3 X^3  0 X^3 X^3  0  0  0 X^3 X^3  0  0  0 X^3 X^3  0  0 X^3  0  0  0
 0  0  0 X^3  0 X^3 X^3 X^3 X^3  0 X^3  0 X^3  0 X^3  0 X^3  0  0 X^3  0 X^3  0 X^3  0  0 X^3 X^3 X^3  0 X^3  0  0  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  0 X^3 X^3 X^3  0 X^3 X^3 X^3  0 X^3  0  0 X^3  0  0 X^3  0 X^3 X^3 X^3
 0  0  0  0 X^3  0 X^3 X^3 X^3 X^3  0 X^3  0 X^3  0  0 X^3  0 X^3  0 X^3 X^3  0 X^3 X^3  0 X^3  0 X^3  0 X^3 X^3 X^3  0  0  0 X^3  0  0 X^3  0 X^3  0 X^3  0 X^3  0 X^3  0 X^3  0 X^3  0 X^3 X^3  0 X^3  0  0  0  0 X^3 X^3 X^3  0  0  0  0 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+216x^66+128x^67+182x^68+128x^69+736x^70+128x^71+182x^72+128x^73+216x^74+1x^76+1x^100+1x^104

The gray image is a linear code over GF(2) with n=560, k=11 and d=264.
This code was found by Heurico 1.16 in 0.25 seconds.