The generator matrix

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

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

Homogenous weight enumerator: w(x)=1x^0+288x^64+1322x^65+2636x^66+3920x^67+5806x^68+7102x^69+7663x^70+8304x^71+8102x^72+7124x^73+5408x^74+3534x^75+2178x^76+1198x^77+556x^78+204x^79+101x^80+54x^81+24x^82+6x^83+4x^84+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 38.4 seconds.