The generator matrix

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

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+308x^81+271x^82+448x^83+302x^84+648x^85+450x^86+512x^87+267x^88+336x^89+180x^90+164x^91+34x^92+72x^93+22x^94+56x^95+3x^96+12x^97+5x^98+4x^99+1x^136

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 103 seconds.