The generator matrix

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

generates a code of length 77 over Z4[X]/(X^2+2) who´s minimum homogenous weight is 72.

Homogenous weight enumerator: w(x)=1x^0+138x^72+668x^73+786x^74+1378x^75+1124x^76+1070x^77+762x^78+764x^79+392x^80+360x^81+232x^82+258x^83+115x^84+94x^85+22x^86+16x^87+5x^88+6x^90+1x^92

The gray image is a code over GF(2) with n=616, k=13 and d=288.
This code was found by Heurico 1.16 in 1.05 seconds.