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

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

Homogenous weight enumerator: w(x)=1x^0+66x^84+128x^86+49x^88+6x^92+5x^96+1x^120

The gray image is a code over GF(2) with n=344, k=8 and d=168.
This code was found by Heurico 1.16 in 14.9 seconds.