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 2X  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 2X+2  1  1  X  1  X
 0  X  0  X  0  0  X  X 2X+2 3X+2 2X+2 3X+2 2X+2 2X+2 3X+2 3X+2  0  0  X  X  0  0  X  X 2X+2 2X+2 2X 3X+2 3X+2 2X+2 2X+2 3X+2 3X+2 2X 2X 2X 3X 3X 2X 2X 3X 3X  2  2 X+2 X+2  2  2 X+2 X+2 2X 2X 3X 3X 2X 2X 3X 3X  2  2 X+2 X+2  2  2 X+2 X+2  0  0  X  X  0 2X+2  2 X+2 X+2  X X+2  X  X X+2  X  2  0 X+2 2X+2 3X
 0  0  X  X  2 3X+2 X+2 2X+2 2X+2 3X+2 3X 2X X+2 2X 3X  2 2X X+2 3X  2 3X 2X+2 3X+2 2X  2  X  X X+2  0  0 3X+2  X 2X+2  X 2X X+2 3X  2 2X+2  X 3X+2  0  2 3X X+2 2X  0 3X+2  X 2X+2  0 3X+2  X 2X+2  2 3X X+2 2X 2X+2  X 3X+2  0 2X X+2 3X  2  0 3X+2  X 2X+2  2 3X 3X+2 3X 2X X+2 2X+2  0 2X  X 3X  2 3X 3X 2X 2X

generates a code of length 86 over Z4[X]/(X^2+2X+2) 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 0.469 seconds.