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

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+282x^66+32x^67+230x^68+224x^69+538x^70+224x^71+206x^72+32x^73+254x^74+9x^76+14x^78+1x^80+1x^124

The gray image is a linear code over GF(2) with n=560, k=11 and d=264.
This code was found by Heurico 1.16 in 11 seconds.