The generator matrix

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

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+164x^78+684x^79+572x^80+808x^81+386x^82+480x^83+236x^84+288x^85+90x^86+100x^87+94x^88+104x^89+53x^90+32x^91+1x^96+3x^98

The gray image is a code over GF(2) with n=656, k=12 and d=312.
This code was found by Heurico 1.16 in 0.406 seconds.