The generator matrix

 1  0  0  1  1  1  2  2 2X+2  1  1  2  1  1 3X  1  1 X+2 3X+2  1  1  1  X  1  1  1  0 2X 3X+2  1  2  1  1  1 X+2  X  1  1  1  1  1  1  1  1  1 3X 2X+2  1  1 2X  1  1 X+2  X  1  1  X  X 2X  1  0 3X+2  1  1  1  1  1  1  1  1 2X+2  1 2X+2 2X+2  1  1  1  X  2  1  0 3X  2 3X  1  1
 0  1  0  0 2X+3 2X+3  1 3X  1 2X  3  1  2 2X+1 3X+2  X 3X+1  1  1 3X X+1 3X+2  1 X+2 X+3 X+3 2X+2  1  0 X+1  1 2X+2 2X  X  1  1  0 2X+2 2X+1 3X+3 2X+2 2X+3 3X+3  X 3X+3  1 X+2  3  X  1  3 X+1 2X  1  3 X+1 3X  1  X  1  1  1  0  X X+3 2X+3 2X+1 X+2 3X+3  2  1  3  0  1 X+2 X+2 X+1  2  0 2X+3  1  1  1 3X+2  3 2X+2
 0  0  1 X+1 3X+1 2X X+3  1  X 3X  X  3 2X+3  3  1 2X+1 3X X+3 X+2  2  3 X+2  0 3X+3 2X X+3  1  1  1 2X 3X+1 2X+1 2X+2 2X+3 2X+1  0  X  2 2X 2X+1 3X+3  1  X  2 3X+1 X+2  1  2 X+1 2X X+2 3X+1  1 2X+3 3X+2  1  1 X+3  1 X+1 3X  X 3X+3 X+2 2X X+3  3  X 2X+3 3X+1 2X  3  1  X  2 2X+2  3  1  1 2X+1  2 X+1 3X+1  1 3X+1 3X+2
 0  0  0 2X 2X  0 2X 2X 2X 2X 2X  0  0  0  0 2X  0  0  0 2X 2X  0 2X  0 2X  0 2X 2X  0  0  0 2X  0  0  0  0  0 2X 2X  0  0  0 2X  0  0 2X  0  0 2X 2X 2X 2X 2X 2X  0  0 2X 2X  0 2X  0 2X 2X  0  0  0 2X 2X 2X  0 2X 2X  0  0  0 2X  0 2X 2X 2X  0  0 2X 2X  0 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+164x^81+700x^82+1078x^83+1127x^84+1016x^85+1011x^86+900x^87+537x^88+380x^89+478x^90+278x^91+205x^92+168x^93+61x^94+36x^95+32x^96+8x^97+4x^98+4x^99+2x^100+1x^102+1x^110

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 1.33 seconds.