The generator matrix

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

generates a code of length 79 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 72.

Homogenous weight enumerator: w(x)=1x^0+346x^72+1652x^73+2856x^74+4312x^75+5780x^76+7070x^77+7371x^78+7896x^79+7266x^80+6556x^81+5290x^82+4108x^83+2393x^84+1474x^85+603x^86+276x^87+159x^88+40x^89+44x^90+16x^91+15x^92+8x^93+4x^94

The gray image is a code over GF(2) with n=632, k=16 and d=288.
This code was found by Heurico 1.16 in 48.7 seconds.