The generator matrix

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

generates a code of length 70 over Z2[X]/(X^4) who´s minimum homogenous weight is 65.

Homogenous weight enumerator: w(x)=1x^0+118x^65+628x^66+1010x^67+1304x^68+740x^69+1138x^70+874x^71+836x^72+530x^73+451x^74+190x^75+177x^76+80x^77+45x^78+38x^79+25x^80+4x^81+1x^82+1x^84+1x^86

The gray image is a linear code over GF(2) with n=560, k=13 and d=260.
This code was found by Heurico 1.16 in 3.8 seconds.