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

generates a code of length 87 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 82.

Homogenous weight enumerator: w(x)=1x^0+178x^82+790x^83+941x^84+1044x^85+1219x^86+952x^87+680x^88+752x^89+512x^90+334x^91+235x^92+256x^93+125x^94+48x^95+60x^96+44x^97+12x^98+4x^99+1x^100+2x^102+1x^104+1x^108

The gray image is a code over GF(2) with n=696, k=13 and d=328.
This code was found by Heurico 1.16 in 1.36 seconds.