The generator matrix

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

generates a code of length 86 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 81.

Homogenous weight enumerator: w(x)=1x^0+118x^81+384x^82+458x^83+542x^84+390x^85+549x^86+326x^87+382x^88+362x^89+362x^90+104x^91+68x^92+24x^93+6x^94+6x^95+6x^96+2x^101+2x^106+2x^107+1x^118+1x^120

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 0.797 seconds.