The generator matrix

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

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+113x^72+728x^73+1342x^74+2152x^75+2902x^76+3864x^77+3794x^78+4072x^79+3715x^80+3144x^81+2444x^82+2004x^83+1120x^84+628x^85+334x^86+232x^87+69x^88+48x^89+18x^90+20x^91+14x^92+4x^93+4x^94+2x^96

The gray image is a code over GF(2) with n=632, k=15 and d=288.
This code was found by Heurico 1.16 in 12.3 seconds.