The generator matrix

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

generates a code of length 71 over Z2[X]/(X^4) who´s minimum homogenous weight is 64.

Homogenous weight enumerator: w(x)=1x^0+60x^64+488x^65+1358x^66+2124x^67+3057x^68+3692x^69+3612x^70+4324x^71+4019x^72+3464x^73+2596x^74+1852x^75+1024x^76+572x^77+292x^78+108x^79+67x^80+8x^81+30x^82+8x^83+11x^84+1x^88

The gray image is a linear code over GF(2) with n=568, k=15 and d=256.
This code was found by Heurico 1.16 in 14.8 seconds.