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

generates a code of length 77 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 72.

Homogenous weight enumerator: w(x)=1x^0+31x^72+204x^73+235x^74+248x^75+219x^76+328x^77+163x^78+216x^79+92x^80+92x^81+144x^82+64x^83+9x^84+1x^86+1x^130

The gray image is a code over GF(2) with n=616, k=11 and d=288.
This code was found by Heurico 1.16 in 0.547 seconds.