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

generates a code of length 86 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 80.

Homogenous weight enumerator: w(x)=1x^0+67x^80+240x^81+287x^82+338x^83+412x^84+560x^85+421x^86+536x^87+429x^88+256x^89+203x^90+182x^91+54x^92+28x^93+23x^94+24x^95+20x^96+4x^97+2x^98+8x^99+1x^144

The gray image is a code over GF(2) with n=688, k=12 and d=320.
This code was found by Heurico 1.16 in 1.08 seconds.