The generator matrix

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

generates a code of length 64 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 56.

Homogenous weight enumerator: w(x)=1x^0+68x^56+692x^57+2141x^58+4402x^59+7492x^60+10526x^61+14198x^62+17226x^63+17512x^64+17600x^65+14704x^66+10504x^67+6746x^68+3866x^69+2033x^70+810x^71+331x^72+144x^73+41x^74+18x^75+8x^76+4x^77+3x^78+2x^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 150 seconds.