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

generates a code of length 80 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 77.

Homogenous weight enumerator: w(x)=1x^0+24x^77+288x^78+40x^79+361x^80+40x^81+200x^82+24x^83+35x^84+8x^86+2x^88+1x^132

The gray image is a code over GF(2) with n=640, k=10 and d=308.
This code was found by Heurico 1.16 in 0.36 seconds.