The generator matrix

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

generates a code of length 71 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 67.

Homogenous weight enumerator: w(x)=1x^0+128x^67+576x^68+748x^69+682x^70+560x^71+331x^72+328x^73+208x^74+192x^75+147x^76+76x^77+78x^78+16x^79+22x^80+2x^84+1x^92

The gray image is a code over GF(2) with n=568, k=12 and d=268.
This code was found by Heurico 1.16 in 0.281 seconds.