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  1  1  1  1  1  1  1  1  1  1  1  X  X
 0  X  0 X^2+X X^2 X^3+X^2+X X^3+X^2  X  0 X^2+X  0 X^3+X^2+X X^2  X X^3+X^2  X  0 X^2+X  0 X^3+X^2+X  0 X^2+X  0 X^3+X^2+X X^2  X X^3+X^2  X X^2  X X^3+X^2  X X^3 X^3+X^2+X X^3 X^2+X X^3+X^2 X^3+X X^2 X^3+X X^3 X^3+X^2+X X^3 X^2+X X^3+X^2 X^3+X X^2 X^3+X X^3 X^3+X^2+X X^2 X^3+X X^3 X^2+X X^3+X^2 X^3+X X^3+X^2+X X^3 X^2 X^3+X X^3 X^2+X X^3+X^2 X^3+X  0 X^2+X X^3+X^2 X^3+X X^3+X^2+X X^3+X^2+X
 0  0 X^3+X^2  0 X^3+X^2 X^2  0 X^2 X^3 X^3 X^2 X^3+X^2 X^2 X^3+X^2 X^3 X^3  0  0 X^3+X^2 X^2 X^3 X^3 X^2 X^3+X^2 X^2 X^2  0 X^3 X^3+X^2 X^3+X^2 X^3  0 X^3 X^3 X^2 X^3+X^2 X^3+X^2 X^2  0  0  0  0 X^3+X^2 X^2 X^2 X^3+X^2 X^3 X^3 X^3 X^3 X^3 X^3 X^2 X^3+X^2 X^2 X^2  0  0  0  0 X^3+X^2 X^2 X^3+X^2 X^3+X^2  0  0  0 X^3 X^2 X^3+X^2
 0  0  0 X^3 X^3  0 X^3 X^3  0 X^3  0  0 X^3 X^3 X^3  0 X^3  0 X^3 X^3 X^3  0 X^3 X^3  0  0  0 X^3  0  0  0 X^3  0  0  0  0 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 X^3 X^3  0  0 X^3 X^3  0  0 X^3 X^3  0  0  0  0 X^3 X^3  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+87x^68+128x^69+592x^70+128x^71+86x^72+1x^76+1x^136

The gray image is a linear code over GF(2) with n=560, k=10 and d=272.
This code was found by Heurico 1.16 in 0.453 seconds.