The generator matrix

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

generates a code of length 77 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 70.

Homogenous weight enumerator: w(x)=1x^0+306x^70+1510x^71+2871x^72+4484x^73+5708x^74+6644x^75+7707x^76+7922x^77+7520x^78+6606x^79+5583x^80+3796x^81+2398x^82+1394x^83+512x^84+338x^85+122x^86+44x^87+29x^88+20x^89+10x^90+10x^91+1x^92

The gray image is a code over GF(2) with n=616, k=16 and d=280.
This code was found by Heurico 1.16 in 48.5 seconds.