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

generates a code of length 71 over Z2[X]/(X^4) who´s minimum homogenous weight is 70.

Homogenous weight enumerator: w(x)=1x^0+80x^70+128x^71+30x^72+8x^74+4x^78+1x^80+4x^86

The gray image is a linear code over GF(2) with n=568, k=8 and d=280.
This code was found by Heurico 1.16 in 1.2 seconds.