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  X  X  X  X  X  X  X  X  1  1  X  X  X  X  1  X  X  1  X  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  X  1  1  1  1  1  1
 0 X^3  0  0  0  0  0  0  0 X^3 X^3 X^3 X^3 X^3 X^3 X^3  0  0  0  0  0  0  0  0 X^3 X^3 X^3 X^3 X^3 X^3 X^3 X^3 X^3 X^3 X^3 X^3  0  0  0  0 X^3  0  0 X^3  0 X^3 X^3  0  0  0  0  0 X^3 X^3  0 X^3 X^3  0  0 X^3 X^3 X^3 X^3  0  0  0  0  0 X^3 X^3
 0  0 X^3  0  0  0 X^3 X^3 X^3 X^3 X^3  0 X^3 X^3  0  0  0  0  0  0 X^3 X^3 X^3 X^3 X^3 X^3 X^3 X^3 X^3 X^3  0  0  0  0 X^3 X^3 X^3 X^3 X^3 X^3  0  0  0  0  0  0  0  0  0  0  0 X^3 X^3  0 X^3 X^3  0 X^3 X^3 X^3 X^3  0  0  0  0  0  0 X^3 X^3  0
 0  0  0 X^3  0 X^3 X^3 X^3  0  0  0  0 X^3 X^3 X^3 X^3  0  0 X^3 X^3 X^3 X^3  0  0  0  0  0  0 X^3 X^3 X^3 X^3  0  0 X^3 X^3 X^3 X^3  0  0 X^3  0 X^3  0 X^3 X^3  0  0  0 X^3 X^3 X^3  0 X^3 X^3 X^3  0  0  0  0 X^3 X^3  0  0  0  0 X^3 X^3  0  0
 0  0  0  0 X^3 X^3  0 X^3 X^3  0 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  0 X^3 X^3  0  0 X^3 X^3  0 X^3  0  0 X^3 X^3  0  0 X^3 X^3  0  0 X^3 X^3  0 X^3 X^3  0  0  0  0 X^3 X^3  0 X^3  0 X^3  0 X^3 X^3  0  0 X^3 X^3  0  0  0

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+15x^68+222x^70+15x^72+2x^86+1x^108

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 0.203 seconds.