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

generates a code of length 76 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 72.

Homogenous weight enumerator: w(x)=1x^0+130x^72+766x^73+522x^74+690x^75+542x^76+394x^77+262x^78+294x^79+98x^80+172x^81+86x^82+96x^83+19x^84+20x^85+1x^88+2x^90+1x^92

The gray image is a code over GF(2) with n=608, k=12 and d=288.
This code was found by Heurico 1.16 in 0.344 seconds.