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  1  1  1  1 X+2  1  2  1 X^2
 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 X^2 X+3  2  X  1  X  1  2  X
 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  2  0  2 X^2+X+2  2  2  2  X X^2+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+412x^78+236x^79+394x^80+92x^81+406x^82+148x^83+227x^84+20x^85+72x^86+16x^87+12x^88+2x^90+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 0.422 seconds.