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