The generator matrix

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

generates a code of length 82 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 78.

Homogenous weight enumerator: w(x)=1x^0+54x^78+330x^79+281x^80+386x^81+193x^82+172x^83+221x^84+260x^85+68x^86+46x^87+8x^88+14x^89+4x^90+4x^93+4x^95+1x^114+1x^116

The gray image is a code over GF(2) with n=656, k=11 and d=312.
This code was found by Heurico 1.16 in 0.422 seconds.