The generator matrix

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

generates a code of length 71 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 64.

Homogenous weight enumerator: w(x)=1x^0+224x^64+1328x^65+2576x^66+4214x^67+5793x^68+6790x^69+7744x^70+8728x^71+7708x^72+6974x^73+5453x^74+3840x^75+2034x^76+1150x^77+591x^78+200x^79+103x^80+42x^81+19x^82+10x^83+9x^84+4x^85+1x^86

The gray image is a code over GF(2) with n=568, k=16 and d=256.
This code was found by Heurico 1.16 in 41.7 seconds.