The generator matrix

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

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

Homogenous weight enumerator: w(x)=1x^0+334x^78+320x^79+383x^80+168x^81+294x^82+248x^83+178x^84+24x^85+50x^86+8x^87+28x^88+2x^90+8x^94+1x^108+1x^124

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 0.453 seconds.