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  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  0  1  1  X  1  1  1  1  1  1  X  1  1 X^2+2  X  1  0
 0  X  0  X  2  2 X+2 X+2 X^2 X^2+X X^2 X^2+X+2 X^2+2 X^2+X X^2+2 X^2+X+2 X^2+X+2  2 X^2 X^2+X+2 X^2+X+2 X^2+X+2  0 X^2 X^2+2  0  X  X X^2+2  X  2 X+2  0  X X^2+X X^2+2 X^2+X+2  2 X+2 X^2+2  2 X^2+X  2 X^2+X+2  X X^2  X  2 X^2 X^2+X X^2+X+2  X X^2+2 X^2+2 X+2 X^2+X  2 X+2 X^2+2 X^2+2  0  X  0  0 X^2+X+2 X^2 X^2+X+2 X^2+2  2  X  0 X^2+X  0 X^2+X  X  2 X^2+X X^2+X  0  X X+2 X+2  0 X^2+2 X^2  0
 0  0  X  X X^2 X^2+X+2 X^2+X X^2+2 X^2 X^2+X  X X^2+2 X^2+X+2  0  0 X+2  X X^2 X^2+X X^2 X^2+X  2  0 X+2  2 X^2+X X^2+2 X+2 X^2+2 X^2+X X+2  0 X^2+2 X^2+X+2  2 X+2 X+2 X^2+X+2 X^2  2  X X^2+2  0 X^2+X+2  0 X^2+X X+2  0 X^2 X+2 X^2+X  2 X^2+X  0 X^2 X^2 X+2 X^2+X+2  X X^2+2 X^2+2  X X^2+X X+2  0 X+2  2 X+2 X^2+X  2 X^2  0  2  X X^2 X^2+2 X^2+X X^2+X+2 X^2 X+2 X+2 X^2+X+2  X X+2 X^2+X  X
 0  0  0  2  2  2  0  2  0  2  2  0  0  2  2  0  2  0  2  2  0  0  2  0  0  0  0  0  2  2  2  2  0  0  2  2  2  0  0  2  2  0  2  0  2  0  2  0  2  0  2  0  2  0  2  2  0  2  0  0  2  0  2  0  0  2  2  0  2  0  2  0  2  2  2  0  0  2  0  0  2  0  0  2  0  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+170x^82+64x^83+262x^84+396x^85+360x^86+388x^87+178x^88+28x^89+126x^90+12x^91+38x^92+8x^93+16x^94+1x^160

The gray image is a code over GF(2) with n=688, k=11 and d=328.
This code was found by Heurico 1.16 in 0.797 seconds.