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  1  1  1  1  1  1  1  1  1  2  0  1  X  1  1 X^2+2  1 X^2  1  1
 0  X X^2+2 X^2+X  0 X^2+X X^2+2 X+2  0 X^2+X X+2 X^2+2  2 X^2+X+2 X^2 X+2  0 X^2+X X+2 X^2+2 X^2+X  0 X^2+2 X+2 X^2+X  0 X+2 X^2+2 X^2+X+2  2 X^2  X  0 X^2+X X^2+2 X+2 X^2+X+2  0 X^2  X X^2+X  2 X+2 X^2 X^2+X+2  2 X^2+2  X  0  2  0 X^2+X X^2+X+2 X^2+X  2 X^2+X+2 X^2+2 X^2+2 X^2 X^2 X+2 X+2  X  X  0  0  2  2 X^2+X X^2+X X^2+X+2 X^2+X+2  2  X  0 X^2+X  2 X^2+2  X X+2 X^2+2 X^2+X X^2+X
 0  0  2  0  0  0  2  0  0  0  0  2  0  0  2  0  0  2  2  0  2  0  0  2  2  2  2  2  2  2  2  2  0  0  0  0  2  2  2  2  0  2  0  2  2  0  0  2  2  2  2  0  2  2  2  0  0  0  0  0  0  2  2  0  0  0  0  0  0  0  0  0  0  2  2  2  2  2  2  2  2  2  2
 0  0  0  2  0  0  0  2  0  0  0  0  0  2  0  2  2  2  2  2  2  2  2  2  0  2  0  2  0  2  2  0  2  0  0  2  2  0  2  2  0  0  2  2  2  2  0  2  2  0  2  2  0  0  0  2  0  2  2  0  0  0  0  0  0  2  2  0  2  2  2  2  0  2  2  2  2  2  2  2  2  0  2
 0  0  0  0  2  0  2  0  0  2  2  2  2  2  0  2  2  0  0  2  2  0  0  2  2  0  2  0  0  2  2  0  2  0  0  2  2  0  2  2  2  2  0  0  0  0  2  0  0  2  2  2  0  2  0  0  0  0  2  2  0  2  0  2  2  0  0  2  2  0  0  2  0  0  2  2  0  0  0  0  0  2  2
 0  0  0  0  0  2  0  2  2  2  0  2  2  0  2  2  0  0  2  2  0  2  0  2  0  0  2  2  0  2  0  2  2  0  0  0  2  0  2  0  0  2  0  0  2  0  2  0  2  0  0  2  2  2  2  2  2  2  0  0  2  0  0  2  2  0  2  0  0  2  0  2  0  0  2  2  2  0  2  0  2  0  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+57x^78+64x^79+198x^80+256x^81+306x^82+432x^83+191x^84+272x^85+66x^86+80x^87+57x^88+48x^89+18x^90+1x^94+1x^156

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 0.781 seconds.