The generator matrix

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

generates a code of length 64 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 56.

Homogenous weight enumerator: w(x)=1x^0+100x^56+920x^57+2397x^58+4616x^59+6823x^60+11016x^61+13894x^62+16660x^63+17328x^64+17960x^65+14157x^66+10980x^67+6692x^68+4180x^69+1912x^70+804x^71+361x^72+160x^73+52x^74+28x^75+15x^76+4x^77+4x^78+8x^80

The gray image is a code over GF(2) with n=512, k=17 and d=224.
This code was found by Heurico 1.16 in 138 seconds.