The generator matrix

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

generates a code of length 55 over Z4[X]/(X^2+2,2X) who´s minimum homogenous weight is 44.

Homogenous weight enumerator: w(x)=1x^0+77x^44+392x^45+1000x^46+1654x^47+2728x^48+3954x^49+5815x^50+7956x^51+9660x^52+11824x^53+13341x^54+13598x^55+13380x^56+12442x^57+10263x^58+8080x^59+5755x^60+3578x^61+2417x^62+1582x^63+839x^64+372x^65+210x^66+84x^67+40x^68+14x^69+10x^70+6x^71

The gray image is a code over GF(2) with n=220, k=17 and d=88.
This code was found by Heurico 1.13 in 188 seconds.