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

generates a code of length 70 over Z2[X]/(X^4) who´s minimum homogenous weight is 67.

Homogenous weight enumerator: w(x)=1x^0+16x^67+55x^68+112x^69+656x^70+112x^71+54x^72+16x^73+1x^76+1x^136

The gray image is a linear code over GF(2) with n=560, k=10 and d=268.
This code was found by Heurico 1.16 in 18 seconds.