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

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

Homogenous weight enumerator: w(x)=1x^0+27x^50+186x^51+118x^52+96x^53+502x^54+210x^55+494x^56+96x^57+100x^58+174x^59+26x^60+10x^62+6x^63+1x^64+1x^98

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 0.203 seconds.