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

generates a code of length 70 over Z2[X]/(X^4) who´s minimum homogenous weight is 69.

Homogenous weight enumerator: w(x)=1x^0+30x^69+206x^70+14x^72+1x^78+1x^80+2x^85+1x^94

The gray image is a linear code over GF(2) with n=560, k=8 and d=276.
This code was found by Heurico 1.16 in 0.344 seconds.