The generator matrix

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

generates a code of length 78 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 71.

Homogenous weight enumerator: w(x)=1x^0+424x^71+1421x^72+3036x^73+4318x^74+5564x^75+7017x^76+7196x^77+8219x^78+7556x^79+6809x^80+5062x^81+3846x^82+2502x^83+1265x^84+776x^85+295x^86+104x^87+61x^88+26x^89+26x^90+10x^91+2x^96

The gray image is a code over GF(2) with n=624, k=16 and d=284.
This code was found by Heurico 1.16 in 46.4 seconds.