The generator matrix

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

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

Homogenous weight enumerator: w(x)=1x^0+106x^72+780x^73+748x^74+1418x^75+891x^76+1206x^77+687x^78+834x^79+424x^80+414x^81+151x^82+304x^83+89x^84+96x^85+26x^86+4x^87+5x^88+3x^90+4x^92+1x^94

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 1.09 seconds.