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

generates a code of length 70 over Z2[X]/(X^4) who´s minimum homogenous weight is 66.

Homogenous weight enumerator: w(x)=1x^0+16x^66+55x^68+880x^70+55x^72+16x^74+1x^140

The gray image is a linear code over GF(2) with n=560, k=10 and d=264.
This code was found by Heurico 1.16 in 0.359 seconds.