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

generates a code of length 78 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 73.

Homogenous weight enumerator: w(x)=1x^0+104x^73+750x^74+832x^75+1518x^76+724x^77+1226x^78+672x^79+760x^80+436x^81+426x^82+208x^83+340x^84+76x^85+78x^86+16x^87+18x^88+4x^89+2x^92+1x^96

The gray image is a code over GF(2) with n=624, k=13 and d=292.
This code was found by Heurico 1.16 in 1.09 seconds.