The generator matrix

 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  X  1  1  1  1  1  1  1  1  0  1  1  1  1  1  1  1  0  1 X^2  1  1  1  0  X  1  X  X  1  1  1  1  X  0 X^2  1 X^2+2  1  1  2  X  1  1  X  0  X X^2+2  1  1
 0  X  0  X  2  0 X^2+X X^2+X+2  0  2 X+2 X+2  0 X^2+X+2 X^2+2  X X^2+2 X^2+X X^2+X X^2+2 X^2+X+2 X+2 X^2+2 X^2 X^2+2 X^2+X X+2  2  0 X+2 X^2+X+2  X X^2+2 X^2  2  X  2 X^2+X  X  0 X+2 X+2 X^2+X+2  0 X^2 X^2+X  0 X^2  X X^2+X  0 X^2  X X^2+X X+2  X  0 X^2+X+2 X^2+X X+2  X X^2+2 X^2+2  X  2  0 X^2+2  2  0 X^2+2 X^2+2  2 X+2  X  X X^2 X+2  0
 0  0  X  X  0 X^2+X+2 X^2+X  2 X^2 X^2+X+2 X^2+X+2 X^2 X^2+2 X^2  X  X X^2+X+2 X+2  0  2  X  0 X+2 X^2 X^2  X X^2 X^2+X+2 X^2+2 X^2+2 X^2 X^2+X X^2+X  0 X^2+X+2  2  X X^2+X X^2 X^2+2 X^2+X  0 X+2  X  0  2  X  2 X^2+X+2 X^2  0 X^2+X+2 X^2+X  X X^2+X+2  2  X X^2+X X^2+X+2 X^2+X+2 X^2+X  X  X X^2 X^2+X  X X+2 X+2  X X+2  0 X+2 X^2+2 X^2+X+2 X^2  X X^2+2  0
 0  0  0 X^2 X^2+2 X^2  2 X^2 X^2  0 X^2 X^2+2  0  0 X^2+2  2 X^2 X^2+2  0 X^2  0 X^2  0  0 X^2  2  2 X^2+2  2 X^2 X^2 X^2+2  2  2  2  2 X^2+2 X^2 X^2+2 X^2+2  2 X^2+2 X^2  2  0  2  2 X^2+2 X^2+2  2  2 X^2+2  0 X^2  0 X^2+2 X^2+2  0 X^2+2 X^2+2  2  2  0  2 X^2+2 X^2+2 X^2  2 X^2 X^2+2 X^2 X^2+2  2  2  0  2  0  0

generates a code of length 78 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 72.

Homogenous weight enumerator: w(x)=1x^0+47x^72+248x^73+269x^74+356x^75+457x^76+544x^77+485x^78+504x^79+386x^80+288x^81+144x^82+132x^83+79x^84+64x^85+21x^86+32x^87+25x^88+8x^89+1x^90+4x^92+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 0.828 seconds.