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

generates a code of length 86 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 85.

Homogenous weight enumerator: w(x)=1x^0+72x^85+143x^86+8x^87+15x^88+12x^89+4x^101+1x^110

The gray image is a code over GF(2) with n=688, k=8 and d=340.
This code was found by Heurico 1.16 in 3.48 seconds.