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

generates a code of length 71 over Z2[X]/(X^4) who´s minimum homogenous weight is 68.

Homogenous weight enumerator: w(x)=1x^0+24x^68+52x^69+100x^70+668x^71+109x^72+44x^73+19x^74+4x^75+2x^76+1x^138

The gray image is a linear code over GF(2) with n=568, k=10 and d=272.
This code was found by Heurico 1.16 in 19.4 seconds.