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

generates a code of length 83 over Z4[X]/(X^2+2,2X) who´s minimum homogenous weight is 84.

Homogenous weight enumerator: w(x)=1x^0+3x^84+16x^85+3x^86+8x^87+1x^90

The gray image is a code over GF(2) with n=332, k=5 and d=168.
This code was found by Heurico 1.16 in 0.29 seconds.