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

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

Homogenous weight enumerator: w(x)=1x^0+24x^86+5x^88+2x^92

The gray image is a code over GF(2) with n=672, k=5 and d=344.
This code was found by Heurico 1.16 in 0.391 seconds.