The generator matrix

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

generates a code of length 86 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 82.

Homogenous weight enumerator: w(x)=1x^0+180x^82+708x^83+660x^84+604x^85+506x^86+410x^87+268x^88+232x^89+130x^90+118x^91+69x^92+96x^93+54x^94+40x^95+17x^96+1x^102+1x^104+1x^106

The gray image is a code over GF(2) with n=688, k=12 and d=328.
This code was found by Heurico 1.16 in 0.453 seconds.