The generator matrix

 1  0  0  1  1  1  2 2X  0 2X+2  1  1  1  1  X  1  1  X  1  1 3X  1  1  1 X+2 3X+2  1 2X 3X 3X  1  1 X+2  X  1  1  1  1  2  1  1 X+2 3X  1  1 3X X+2  1 3X X+2 2X  1  2  2  1  1  1  1  1 X+2  1  0  0  1 2X+2  1 2X  2 3X  X 2X  1  1 2X  1  1  2  1
 0  1  0  0 2X+3  3  1 3X+2  1  1  0 2X 2X+3 2X+3  X 3X+2 3X+3  1  X 3X+1  1 3X+1 3X+2 X+2 2X+2  1 X+1  1 3X+2  1 X+3 2X+2  1  1 2X 2X+1 X+3  X  1 2X+1  2  X  1  2 2X+2  1  1  1  1  1  1 X+1  1 3X+2 2X X+3 X+3  1 3X+2  0 X+1 2X+2 3X  X  1 2X+3  1  0  2  1  1  0 X+2  1 3X+2 X+3  1  0
 0  0  1 X+1 3X+3 2X+2 3X+3  1 X+2  1 X+2  3 2X+1  X  1 2X+3 2X+1 3X 2X 2X+2 2X+1 3X+1  X X+1  1  0 3X 3X+3  1 X+3 3X+2 3X 2X+3 X+2  3  2  3  2 2X+2 3X+3 X+3  1 3X+1 X+1  0 2X+3 X+1 2X+1 2X+2 X+2 X+1 2X+2 2X+1  1 2X+2  X 3X+3 X+1 X+1  1 2X  1  1 3X+1 3X+2 3X X+2  1  1 2X  1  X 2X+1  0 2X 2X+3 X+1  0
 0  0  0 2X+2 2X+2  0 2X+2  2 2X+2 2X  2 2X 2X  2  0  0  0 2X 2X 2X  2  2 2X+2  2  2 2X 2X+2  2 2X  0  0  0 2X+2 2X+2  0  2 2X+2 2X+2  2 2X  0 2X+2 2X  2 2X+2 2X  2  0 2X+2  2 2X  0  0 2X 2X+2  2 2X  0 2X 2X  2 2X+2  0 2X+2 2X 2X  0  2  2  2 2X+2 2X  2 2X 2X 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+104x^72+904x^73+1050x^74+1818x^75+1722x^76+2270x^77+1648x^78+2016x^79+1190x^80+1456x^81+789x^82+682x^83+290x^84+266x^85+82x^86+44x^87+15x^88+14x^89+15x^90+6x^92+2x^97

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 3.69 seconds.