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

generates a code of length 70 over Z2[X]/(X^4) who´s minimum homogenous weight is 68.

Homogenous weight enumerator: w(x)=1x^0+9x^68+64x^69+109x^70+60x^71+6x^72+2x^73+2x^74+2x^85+1x^106

The gray image is a linear code over GF(2) with n=560, k=8 and d=272.
This code was found by Heurico 1.16 in 0.25 seconds.