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

generates a code of length 55 over Z4[X]/(X^2+2) who´s minimum homogenous weight is 50.

Homogenous weight enumerator: w(x)=1x^0+111x^50+4x^51+157x^52+128x^53+400x^54+504x^55+396x^56+128x^57+93x^58+4x^59+69x^60+30x^62+16x^64+6x^66+1x^96

The gray image is a code over GF(2) with n=440, k=11 and d=200.
This code was found by Heurico 1.16 in 21 seconds.