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

generates a code of length 86 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 84.

Homogenous weight enumerator: w(x)=1x^0+256x^84+128x^85+312x^86+128x^87+154x^88+24x^90+20x^92+1x^160

The gray image is a code over GF(2) with n=688, k=10 and d=336.
This code was found by Heurico 1.16 in 0.453 seconds.