The generator matrix

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

generates a code of length 78 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 72.

Homogenous weight enumerator: w(x)=1x^0+25x^72+242x^73+212x^74+530x^75+333x^76+538x^77+416x^78+582x^79+347x^80+410x^81+127x^82+158x^83+58x^84+86x^85+11x^86+10x^87+2x^88+4x^89+1x^90+1x^92+1x^94+1x^120

The gray image is a code over GF(2) with n=624, k=12 and d=288.
This code was found by Heurico 1.16 in 0.563 seconds.