The generator matrix

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

generates a code of length 81 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 78.

Homogenous weight enumerator: w(x)=1x^0+370x^78+268x^79+428x^80+144x^81+300x^82+200x^83+187x^84+16x^85+94x^86+12x^87+18x^88+4x^92+4x^94+1x^108+1x^120

The gray image is a code over GF(2) with n=648, k=11 and d=312.
This code was found by Heurico 1.16 in 1.36 seconds.