The generator matrix

 1  0  0  0  1  1  1  1 X^3  1 X^3+X X^2+X X^3+X^2+X  1  1 X^2+X  1  1  1  X X^3  1  1  1  1 X^3+X X^2+X X^3+X^2  X  1 X^3+X^2+X X^3+X^2+X  1  1  0  1  1 X^3+X^2 X^3+X^2+X  1  1  1  1 X^3+X^2  X X^2  1 X^2 X^3+X^2 X^3+X^2 X^3+X^2  X  1 X^3 X^3  1  1  1  1 X^2+X  1 X^2  0  0  1  0 X^3+X^2  1  1  1
 0  1  0  0  X X^2+1 X^3+X^2+X X^3+X^2+1  1 X^3+X^2+1  1  1 X^3+X  1 X^2+X X^3+X^2+X X^2+X+1 X^3+X^2 X^3+X^2+X  1  1 X+1 X^3+X^2+X X^2+X+1 X^3+X^2+X+1  0 X^3 X^3  1 X^2  1  1  0  1 X^3+X^2 X^2+X X^3+X^2+X+1 X^3+X X^3 X^2+X+1 X^3+X X^3 X^3+X^2+1  1 X^2+X  1 X^3+X^2+X+1  1 X^3+X  1  1  0 X^2 X^2  1 X+1 X^3+X+1 X^3+X^2+X X^3+X+1  1 X^2+1  1 X^2  1 X^2  1  1 X^2+X+1 X^2+1 X^3
 0  0  1  0  0 X^3 X^3+X+1 X+1 X^3+1 X^3+X^2+1 X+1 X^2  1  0 X^3+X^2+1 X^3+X^2+X  1 X^2+X+1  0 X^3 X^2 X^2+X+1 X^3+X X^2+X X^3+X X^2+X  1  1 X^3+X^2+1 X^3+X+1 X^3+X^2+X+1 X+1  X X^3+X^2+X+1  1 X+1 X^2 X^3+X  1 X^3 X^3+X^2+X  X X^3+1 X^3+X^2+X X^3+X^2 X^3 X^3+X+1 X^2+X  1 X+1 X^3+X+1  1 X^2  1  1 X^3+1 X^2 X^2 X^2+1 X^3+X^2+X X^3+X^2 X^2+X+1  1 X^3+X^2+X X^3 X+1 X^3+X+1 X^3+X^2+X+1 X^2+X  0
 0  0  0  1  1 X^3+X+1 X+1 X^3+1  X X^3+X^2 X^3+X^2+X+1 X^3+1 X^3+X+1 X^3+X X^3  1 X^3+X^2+X X^2+1 X^2+X X^2+X X^2+X+1 X^2+X+1  1 X^3+X+1  0  1 X^2 X^3+X^2+1 X^3+X^2+X+1  X X^2+X X^3+1 X^3+X^2+1 X^3+X^2 X+1 X^3+X^2+1  1  1 X^3  X X^2+X X+1 X^3+X^2+X+1 X^3+X^2+X  1  X X^2+X  1 X^3+X X^2+1 X^2 X^2+X X^3+X^2+X  0 X^3 X^2+X+1 X^2 X^3 X^3+X X^3+X^2 X^3+X^2+1 X^3+X^2+1  X X^2+X+1 X^3+X+1 X^3+X X^3+X X^3+X^2+X+1 X^3+X^2+1 X^2
 0  0  0  0 X^3 X^3 X^3 X^3  0  0 X^3 X^3 X^3  0  0 X^3  0 X^3  0  0 X^3 X^3 X^3 X^3  0 X^3  0 X^3  0 X^3 X^3  0  0 X^3  0  0  0  0 X^3 X^3 X^3  0  0 X^3  0 X^3 X^3  0 X^3  0 X^3  0 X^3 X^3 X^3  0 X^3 X^3 X^3 X^3 X^3 X^3 X^3  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 62.

Homogenous weight enumerator: w(x)=1x^0+164x^62+1022x^63+2551x^64+5028x^65+7392x^66+10684x^67+13654x^68+16206x^69+16976x^70+17580x^71+13588x^72+10804x^73+7220x^74+4230x^75+2210x^76+1022x^77+410x^78+206x^79+72x^80+28x^81+10x^82+2x^83+4x^84+2x^86+4x^87+2x^90

The gray image is a linear code over GF(2) with n=560, k=17 and d=248.
This code was found by Heurico 1.16 in 156 seconds.