The generator matrix

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

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+200x^81+438x^82+376x^83+283x^84+512x^85+474x^86+512x^87+290x^88+376x^89+428x^90+200x^91+2x^94+1x^96+2x^114+1x^116

The gray image is a code over GF(2) with n=688, k=12 and d=324.
This code was found by Heurico 1.16 in 0.735 seconds.