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

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+81x^66+120x^67+102x^68+880x^69+54x^70+536x^71+25x^72+32x^73+56x^74+96x^75+64x^76+1x^138

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 1.08 seconds.