The generator matrix

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

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

Homogenous weight enumerator: w(x)=1x^0+206x^81+666x^82+950x^83+1348x^84+940x^85+986x^86+718x^87+760x^88+474x^89+347x^90+214x^91+249x^92+152x^93+110x^94+50x^95+9x^96+4x^97+4x^99+1x^100+2x^102+1x^114

The gray image is a code over GF(2) with n=688, k=13 and d=324.
This code was found by Heurico 1.16 in 1.31 seconds.