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

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

Homogenous weight enumerator: w(x)=1x^0+34x^86+12x^87+3x^88+6x^90+4x^91+4x^92

The gray image is a code over GF(2) with n=688, k=6 and d=344.
This code was found by Heurico 1.16 in 0.438 seconds.