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

generates a code of length 83 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 78.

Homogenous weight enumerator: w(x)=1x^0+36x^78+40x^79+163x^80+248x^81+452x^82+352x^83+350x^84+160x^85+48x^86+56x^87+101x^88+40x^89+1x^160

The gray image is a code over GF(2) with n=664, k=11 and d=312.
This code was found by Heurico 1.16 in 0.703 seconds.