The generator matrix

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

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

Homogenous weight enumerator: w(x)=1x^0+136x^72+700x^73+1264x^74+1792x^75+1786x^76+1894x^77+1881x^78+1956x^79+1543x^80+1208x^81+948x^82+624x^83+276x^84+206x^85+47x^86+60x^87+33x^88+8x^89+16x^90+2x^94+1x^96+2x^98

The gray image is a code over GF(2) with n=624, k=14 and d=288.
This code was found by Heurico 1.16 in 3.61 seconds.