The generator matrix

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

generates a code of length 86 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 81.

Homogenous weight enumerator: w(x)=1x^0+34x^81+93x^82+144x^83+313x^84+270x^85+376x^86+364x^87+215x^88+34x^89+106x^90+36x^91+47x^92+14x^93+1x^162

The gray image is a code over GF(2) with n=688, k=11 and d=324.
This code was found by Heurico 1.16 in 0.813 seconds.