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

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+147x^72+144x^73+229x^74+292x^75+496x^76+440x^77+640x^78+536x^79+460x^80+224x^81+161x^82+132x^83+118x^84+24x^85+16x^86+18x^88+8x^90+4x^92+2x^94+3x^96+1x^128

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 1.22 seconds.