The generator matrix

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

generates a code of length 87 over Z4[X]/(X^2+2,2X) who´s minimum homogenous weight is 86.

Homogenous weight enumerator: w(x)=1x^0+166x^86+42x^88+40x^90+2x^92+1x^96+2x^102+2x^108

The gray image is a code over GF(2) with n=348, k=8 and d=172.
This code was found by Heurico 1.16 in 55.9 seconds.