The generator matrix

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

generates a code of length 70 over Z2[X]/(X^4) who´s minimum homogenous weight is 63.

Homogenous weight enumerator: w(x)=1x^0+208x^63+1182x^64+2608x^65+4192x^66+5536x^67+7138x^68+8368x^69+8189x^70+7792x^71+7036x^72+5332x^73+3716x^74+2122x^75+1194x^76+544x^77+201x^78+92x^79+33x^80+28x^81+14x^82+10x^83

The gray image is a linear code over GF(2) with n=560, k=16 and d=252.
This code was found by Heurico 1.16 in 39.7 seconds.