The generator matrix

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

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+48x^64+676x^65+1150x^66+2234x^67+2728x^68+3776x^69+3658x^70+4826x^71+3541x^72+3572x^73+2494x^74+2046x^75+944x^76+616x^77+204x^78+126x^79+53x^80+32x^81+8x^82+16x^83+12x^84+6x^86+1x^88

The gray image is a code over GF(2) with n=568, k=15 and d=256.
This code was found by Heurico 1.16 in 11.9 seconds.